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<title>Mathieu function</title>

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Mathieu function

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Massieu_function" title="Massieu function">Massieu function</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's <a href="Differential_equation" title="Differential equation">differential equation</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y}{dx^{2}}}+(a-2q\cos(2x))y=0,}</annotation>

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</math><img src="./a8e4d30875926f1274fbee986fa475625fde88eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.251ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}y}{dx^{2}}}+(a-2q\cos(2x))y=0,}" loading="lazy"></dd></dl>

where a, q are <a href="Real_number" title="Real number">real</a>-valued parameters. Since we may add π/2 to x to change the sign of q, it is a usual convention to set q ≥ 0.

They were first introduced by <a href="%C3%89mile_L%C3%A9onard_Mathieu" title="Émile Léonard Mathieu">Émile Léonard Mathieu</a>, who encountered them while studying vibrating elliptical <a href="Drumhead" title="Drumhead">drumheads</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-MorseandFeshbach-2">[2]</a><a href="#cite_note-BrimacombeCorlessZamir-3">[3]</a> They have applications in many fields of the physical sciences, such as <a href="Optics" title="Optics">optics</a>, <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, and <a href="General_relativity" title="General relativity">general relativity</a>. They tend to occur in problems involving periodic motion, or in the analysis of <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a> (PDE) <a href="Boundary_value_problem" title="Boundary value problem">boundary value problems</a> possessing <a href="Elliptic" class="mw-redirect" title="Elliptic">elliptic</a> symmetry.<a href="#cite_note-Gutiérrez-Vega2015-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Mathieu_functions">Mathieu functions</h3></div>

In some usages, Mathieu function refers to solutions of the Mathieu differential equation for arbitrary values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">. When no confusion can arise, other authors use the term to refer specifically to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">- or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">-periodic solutions, which exist only for special values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">.<a href="#cite_note-Arscott_chapIII-5">[5]</a> More precisely, for given (real) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> such periodic solutions exist for an infinite number of values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, called characteristic numbers, conventionally indexed as two separate sequences <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}(q)}">

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<annotation encoding="application/x-tex">{\displaystyle a_{n}(q)}</annotation>

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</math><img src="./d9953e687a1aadc9f45cd28d2a47ef2a36f9c877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.327ex; height:2.843ex;" alt="{\displaystyle a_{n}(q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}(q)}">

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<annotation encoding="application/x-tex">{\displaystyle b_{n}(q)}</annotation>

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</math><img src="./e2dbf776127e1fe1ae4993d06f09b30998ac8390.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.095ex; height:2.843ex;" alt="{\displaystyle b_{n}(q)}" loading="lazy">, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,2,3,\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle n=1,2,3,\ldots }</annotation>

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</math><img src="./a508e267228dcaa193fe3a7df64c94b0f31e143e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\displaystyle n=1,2,3,\ldots }" loading="lazy">. The corresponding functions are denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

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</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

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</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy">, respectively. They are sometimes also referred to as cosine-elliptic and sine-elliptic, or Mathieu functions of the first kind.

As a result of assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is real, both the characteristic numbers and associated functions are real-valued.<a href="#cite_note-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

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</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

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</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy"> can be further classified by <a href="Even_and_odd_functions" title="Even and odd functions">parity</a> and periodicity (both with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">), as follows:<a href="#cite_note-Arscott_chapIII-5">[5]</a>

<th>Function</th>

<th>Parity</th>

<th>Period

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n},\ n{\text{ even}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n},\ n{\text{ odd}}}</annotation>

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</math><img src="./402b2f80a3c2fe41d5db0354c4bcde8e6b0a67c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.621ex; height:2.509ex;" alt="{\displaystyle {\text{ce}}_{n},\ n{\text{ odd}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n},\ n{\text{ even}}}</annotation>

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</math><img src="./92a40a523be7ebc82d4bc7053e0c25beb40ae16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.342ex; height:2.009ex;" alt="{\displaystyle {\text{se}}_{n},\ n{\text{ even}}}" loading="lazy">

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<td>odd

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n},\ n{\text{ odd}}}</annotation>

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</math><img src="./5cb0c60b9f73863ff4480708ea2d1639248c076e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.505ex; height:2.509ex;" alt="{\displaystyle {\text{se}}_{n},\ n{\text{ odd}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">

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The indexing with the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, besides serving to arrange the characteristic numbers in ascending order, is convenient in that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

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</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

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</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy"> become proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos nx}">

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<annotation encoding="application/x-tex">{\displaystyle \cos nx}</annotation>

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</math><img src="./a08b9fef43f2ff87e3d5e980ebe2f635fbee0e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.223ex; height:1.676ex;" alt="{\displaystyle \cos nx}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin nx}">

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<annotation encoding="application/x-tex">{\displaystyle \sin nx}</annotation>

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</math><img src="./fc4cf3dbda60f1aa597014807d61ccfee5be156e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.967ex; height:2.176ex;" alt="{\displaystyle \sin nx}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\rightarrow 0}">

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<annotation encoding="application/x-tex">{\displaystyle q\rightarrow 0}</annotation>

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</math><img src="./9d33351f6c8f42cf0a23c2e26c3dd47ade0b0720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.846ex; height:2.509ex;" alt="{\displaystyle q\rightarrow 0}" loading="lazy">. With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> being an integer, this gives rise to the classification of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}}</annotation>

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</math><img src="./7b3b7459759cb25e6e290ac8deac0d9402fabf51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.283ex; height:2.009ex;" alt="{\displaystyle {\text{ce}}_{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}}</annotation>

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</math><img src="./c1181d702f42d8e7ed3d39cd99b821b3dcab143d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.167ex; height:2.009ex;" alt="{\displaystyle {\text{se}}_{n}}" loading="lazy"> as Mathieu functions (of the first kind) of integral order. For general <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, solutions besides these can be defined, including Mathieu functions of fractional order as well as non-periodic solutions.

<div class="mw-heading mw-heading3"><h3 id="Modified_Mathieu_functions">Modified Mathieu functions</h3></div>

Closely related are the modified Mathieu functions, also known as radial Mathieu functions, which are solutions of Mathieu's modified differential equation

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y}{dx^{2}}}-(a-2q\cosh 2x)y=0,}</annotation>

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</math><img src="./12cdd173f2618a33eec6b6eca7f0b84c1acd2261.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.121ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}y}{dx^{2}}}-(a-2q\cosh 2x)y=0,}" loading="lazy"></dd></dl>

which can be related to the original Mathieu equation by taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \pm {\rm {i}}x}">

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<annotation encoding="application/x-tex">{\displaystyle x\to \pm {\rm {i}}x}</annotation>

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</math><img src="./774ee6b951181fa243e4286715f1504dc071af5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.728ex; height:2.176ex;" alt="{\displaystyle x\to \pm {\rm {i}}x}" loading="lazy">. Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ce}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Ce}}_{n}(x,q)}</annotation>

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</math><img src="./cb3acef7d1afebb040fd9bc472ce7acf3dc162e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.171ex; height:2.843ex;" alt="{\displaystyle {\text{Ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Se}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Se}}_{n}(x,q)}</annotation>

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</math><img src="./61de1cd0160d88eeddbf17249ee35255ac84da75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.786ex; height:2.843ex;" alt="{\displaystyle {\text{Se}}_{n}(x,q)}" loading="lazy">, are defined from<a href="#cite_note-McLachlan_chapII-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{Ce}}_{n}(x,q)&={\text{ce}}_{n}({\rm {i}}x,q).\\{\text{Se}}_{n}(x,q)&=-{\rm {i}}\,{\text{se}}_{n}({\rm {i}}x,q).\end{aligned}}}</annotation>

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</math><img src="./0c4c567408fc313459f17d573eb9405df541a563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.567ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}{\text{Ce}}_{n}(x,q)&={\text{ce}}_{n}({\rm {i}}x,q).\\{\text{Se}}_{n}(x,q)&=-{\rm {i}}\,{\text{se}}_{n}({\rm {i}}x,q).\end{aligned}}}" loading="lazy"></dd></dl>

These functions are real-valued when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is real.

<div class="mw-heading mw-heading3"><h3 id="Normalization">Normalization</h3></div>

A common normalization,<a href="#cite_note-8">[8]</a> which will be adopted throughout this article, is to demand

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{2\pi }{\text{ce}}_{n}(x,q)^{2}dx=\int _{0}^{2\pi }{\text{se}}_{n}(x,q)^{2}dx=\pi }</annotation>

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</math><img src="./e719cfa38f98c13eb359fe552dc4e30d11cdff18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.359ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{2\pi }{\text{ce}}_{n}(x,q)^{2}dx=\int _{0}^{2\pi }{\text{se}}_{n}(x,q)^{2}dx=\pi }" loading="lazy"></dd></dl>

as well as require <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}(x,q)\rightarrow +\cos nx}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)\rightarrow +\cos nx}</annotation>

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</math><img src="./d365e2d6c4f24b58a56878b8694258741919f2a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.558ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)\rightarrow +\cos nx}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)\rightarrow +\sin nx}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)\rightarrow +\sin nx}</annotation>

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</math><img src="./b7f96d8c4d43195148c0561a73a486a289a416a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.186ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)\rightarrow +\sin nx}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\rightarrow 0}">

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<annotation encoding="application/x-tex">{\displaystyle q\rightarrow 0}</annotation>

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</math><img src="./9d33351f6c8f42cf0a23c2e26c3dd47ade0b0720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.846ex; height:2.509ex;" alt="{\displaystyle q\rightarrow 0}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Floquet_theory">Floquet theory</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Floquet_theory" title="Floquet theory">Floquet theory</a></div>

Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called <a href="Floquet_theory" title="Floquet theory">Floquet theory</a>. The central result is Floquet's theorem:

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Floquet's theorem<a href="#cite_note-9">[9]</a>—Mathieu's equation always has at least one solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y(x)}</annotation>

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</math><img src="./e871993bfd131a8b0c3591c26084cf8171a74dcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.294ex; height:2.843ex;" alt="{\displaystyle y(x)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x+\pi )=\sigma y(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y(x+\pi )=\sigma y(x)}</annotation>

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</math><img src="./3e184a129610c10ff312579a8293bb27c37d9d0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.189ex; height:2.843ex;" alt="{\displaystyle y(x+\pi )=\sigma y(x)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is a constant which depends on the parameters of the equation and may be real or complex.

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It is natural to associate the characteristic numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(q)}">

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<annotation encoding="application/x-tex">{\displaystyle a(q)}</annotation>

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</math><img src="./b65d5f7d16bcc43aced61a5fdcb7def3dd8216bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.109ex; height:2.843ex;" alt="{\displaystyle a(q)}" loading="lazy"> with those values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> which result in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =\pm 1}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma =\pm 1}</annotation>

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</math><img src="./ce7b2fac554d7f3e16680e0582a22d551e4ea8f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle \sigma =\pm 1}" loading="lazy">.<a href="#cite_note-10">[10]</a> Note, however, that the theorem only guarantees the existence of at least one solution satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x+\pi )=\sigma y(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y(x+\pi )=\sigma y(x)}</annotation>

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</math><img src="./3e184a129610c10ff312579a8293bb27c37d9d0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.189ex; height:2.843ex;" alt="{\displaystyle y(x+\pi )=\sigma y(x)}" loading="lazy">, when Mathieu's equation in fact has two independent solutions for any given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">. Indeed, it turns out that with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> equal to one of the characteristic numbers, Mathieu's equation has only one periodic solution (that is, with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">), and this solution is one of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

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</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

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</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy">. The other solution is nonperiodic, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{fe}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{fe}}_{n}(x,q)}</annotation>

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</math><img src="./dd63d2f774f9b0773d73ff81eb6161d1104f95da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle {\text{fe}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ge}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ge}}_{n}(x,q)}</annotation>

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</math><img src="./17f21e0b09a619f1dc7ddd8ad33ba0c263215c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.656ex; height:2.843ex;" alt="{\displaystyle {\text{ge}}_{n}(x,q)}" loading="lazy">, respectively, and referred to as a Mathieu function of the second kind.<a href="#cite_note-11">[11]</a> This result can be formally stated as Ince's theorem:

Ince's theorem<a href="#cite_note-12">[12]</a>—Define a basically periodic function as one satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x+\pi )=\pm y(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y(x+\pi )=\pm y(x)}</annotation>

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</math><img src="./251fad74d85a224f420223678e85791f68ffc97d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.668ex; height:2.843ex;" alt="{\displaystyle y(x+\pi )=\pm y(x)}" loading="lazy">. Then, except in the trivial case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=0}">

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<annotation encoding="application/x-tex">{\displaystyle q=0}</annotation>

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</math><img src="./654c2d5dc1a26e0af36dc0deb5fd252c6178977a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=0}" loading="lazy">, Mathieu's equation never possesses two (independent) basically periodic solutions for the same values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">.

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An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form

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<annotation encoding="application/x-tex">{\displaystyle F(a,q,x)=\exp(i\mu \,x)\,P(a,q,x),}</annotation>

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</math><img src="./748944a52acf92827d381519a297ced73ad277f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.914ex; height:2.843ex;" alt="{\displaystyle F(a,q,x)=\exp(i\mu \,x)\,P(a,q,x),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> is a <a href="Complex_number" title="Complex number">complex number</a>, the Floquet exponent (or sometimes Mathieu exponent), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is a complex valued function periodic in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">. An example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(a,q,x)}">

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<annotation encoding="application/x-tex">{\displaystyle P(a,q,x)}</annotation>

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</math><img src="./f5e78047a685c2c59bee9c0b9d4cf1f943dbe035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.252ex; height:2.843ex;" alt="{\displaystyle P(a,q,x)}" loading="lazy"> is plotted to the right.

<div class="mw-heading mw-heading3"><h3 id="Stability_in_parameter_space">Stability in parameter space</h3></div>

The Mathieu equation has two parameters. For almost all choices of these parameters, Floquet theory says that any solution either converges to zero or diverges to infinity.

If the Mathieu equation is parameterized as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {x}}+k(1-m\cos(t))x=0}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\ddot {x}}+k(1-m\cos(t))x=0}</annotation>

</semantics>

</math><img src="./88fb10e13462430baff3ffaee7fca88118fc5ceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.972ex; height:2.843ex;" alt="{\displaystyle {\ddot {x}}+k(1-m\cos(t))x=0}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {R} ,m\geq 0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {R} ,m\geq 0}</annotation>

</semantics>

</math><img src="./0f27af1e6e1576ac206f8aec0243408296098a6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.065ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {R} ,m\geq 0}" loading="lazy">, then the regions of stability and instability are separated by the following curves:<a href="#cite_note-FOOTNOTEButikov2018-13">[13]</a>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(k)={\begin{cases}2{\sqrt {\frac {k(k-1)(k-4)}{3k-8}}},&k<0;\\[4pt]{\frac {1}{4}}\left[{\sqrt {(9-4k)(13-20k)}}-(9-4k)\right],&k<{\frac {1}{4}};\\[10pt]{\frac {1}{4}}\left[9-4k\mp {\sqrt {(9-4k)(13-20k)}}\right],&{\frac {1}{4}}<k<{\frac {13}{20}};\\[6pt]{\sqrt {\frac {2(k-1)(k-4)(k-9)}{k-5}}},&{\frac {13}{20}}<k<1;\\[2pt]2{\sqrt {\frac {k(k-1)(k-4)}{3k-8}}},&k>1.\end{cases}}}">

<mrow>

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<mfrac>

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<mfrac>

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<mfrac>

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<mtd>

<mfrac>

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<mfrac>

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</mtd>

</mtr>

<mtr>

<mtd>

<msqrt>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mtd>

<mtd>

<mfrac>

</mfrac>

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</mtd>

</mtr>

<mtr>

<mtd>

<msqrt>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

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</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m(k)={\begin{cases}2{\sqrt {\frac {k(k-1)(k-4)}{3k-8}}},&k<0;\\[4pt]{\frac {1}{4}}\left[{\sqrt {(9-4k)(13-20k)}}-(9-4k)\right],&k<{\frac {1}{4}};\\[10pt]{\frac {1}{4}}\left[9-4k\mp {\sqrt {(9-4k)(13-20k)}}\right],&{\frac {1}{4}}<k<{\frac {13}{20}};\\[6pt]{\sqrt {\frac {2(k-1)(k-4)(k-9)}{k-5}}},&{\frac {13}{20}}<k<1;\\[2pt]2{\sqrt {\frac {k(k-1)(k-4)}{3k-8}}},&k>1.\end{cases}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Other_types_of_Mathieu_functions">Other types of Mathieu functions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Second_kind">Second kind</h3></div>

Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. Floquet's theory says that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is equal to a characteristic number, one of these solutions can be taken to be periodic, and the other nonperiodic. The periodic solution is one of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy">, called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{fe}}_{n}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{fe}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./dd63d2f774f9b0773d73ff81eb6161d1104f95da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle {\text{fe}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ge}}_{n}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{ge}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./17f21e0b09a619f1dc7ddd8ad33ba0c263215c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.656ex; height:2.843ex;" alt="{\displaystyle {\text{ge}}_{n}(x,q)}" loading="lazy">, respectively, and is called a Mathieu function of the second kind (of integral order). The nonperiodic solutions are unstable, that is, they diverge as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\rightarrow \pm \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\rightarrow \pm \infty }</annotation>

</semantics>

</math><img src="./40b24da34b4a90422e1fb4dc69110723a8bf745e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.834ex; height:2.176ex;" alt="{\displaystyle z\rightarrow \pm \infty }" loading="lazy">.<a href="#cite_note-14">[14]</a>

The second solutions corresponding to the modified Mathieu functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ce}}_{n}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Ce}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./cb3acef7d1afebb040fd9bc472ce7acf3dc162e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.171ex; height:2.843ex;" alt="{\displaystyle {\text{Ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Se}}_{n}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Se}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./61de1cd0160d88eeddbf17249ee35255ac84da75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.786ex; height:2.843ex;" alt="{\displaystyle {\text{Se}}_{n}(x,q)}" loading="lazy"> are naturally defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Fe}}_{n}(x,q)=-i{\text{fe}}_{n}(xi,q)}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Fe}}_{n}(x,q)=-i{\text{fe}}_{n}(xi,q)}</annotation>

</semantics>

</math><img src="./73a46efdb724be6ba32129ec71cd58d7c13ab5ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.728ex; height:2.843ex;" alt="{\displaystyle {\text{Fe}}_{n}(x,q)=-i{\text{fe}}_{n}(xi,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ge}}_{n}(x,q)={\text{ge}}_{n}(xi,q)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Ge}}_{n}(x,q)={\text{ge}}_{n}(xi,q)}</annotation>

</semantics>

</math><img src="./5666a3f2f8c370ce94958e699d1e211c4336a9a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.874ex; height:2.843ex;" alt="{\displaystyle {\text{Ge}}_{n}(x,q)={\text{ge}}_{n}(xi,q)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Fractional_order">Fractional order</h3></div>

Mathieu functions of fractional order can be defined as those solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{p}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{p}(x,q)}</annotation>

</semantics>

</math><img src="./cc117fc64b61cee2ce04c098c91aa695a66642f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.366ex; height:3.009ex;" alt="{\displaystyle {\text{ce}}_{p}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{p}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{p}(x,q)}</annotation>

</semantics>

</math><img src="./7872748dc1b5ae53565bbeae6d1287802adf8288.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.25ex; height:3.009ex;" alt="{\displaystyle {\text{se}}_{p}(x,q)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> a non-integer, which turn into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos px}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos px}</annotation>

</semantics>

</math><img src="./9be819ec8c5f6f8a73d6c4671b2c3960b936fe42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.997ex; height:2.009ex;" alt="{\displaystyle \cos px}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin px}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin px}</annotation>

</semantics>

</math><img src="./283743ba1df0849492c8c436f99ecbc9e9a73d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.742ex; height:2.509ex;" alt="{\displaystyle \sin px}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\rightarrow 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\rightarrow 0}</annotation>

</semantics>

</math><img src="./9d33351f6c8f42cf0a23c2e26c3dd47ade0b0720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.846ex; height:2.509ex;" alt="{\displaystyle q\rightarrow 0}" loading="lazy">.<a href="#cite_note-McLachlan_chapII-7">[7]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is irrational, they are non-periodic; however, they remain bounded as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\rightarrow \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\rightarrow \infty }</annotation>

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</math><img src="./02579e74e2ef1ca0befceba816b311fe5bfd6844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\rightarrow \infty }" loading="lazy">.

An important property of the solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{p}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{p}(x,q)}</annotation>

</semantics>

</math><img src="./cc117fc64b61cee2ce04c098c91aa695a66642f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.366ex; height:3.009ex;" alt="{\displaystyle {\text{ce}}_{p}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{p}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{p}(x,q)}</annotation>

</semantics>

</math><img src="./7872748dc1b5ae53565bbeae6d1287802adf8288.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.25ex; height:3.009ex;" alt="{\displaystyle {\text{se}}_{p}(x,q)}" loading="lazy">, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> non-integer, is that they exist for the same value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">. In contrast, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is an integer, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{p}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{p}(x,q)}</annotation>

</semantics>

</math><img src="./cc117fc64b61cee2ce04c098c91aa695a66642f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.366ex; height:3.009ex;" alt="{\displaystyle {\text{ce}}_{p}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{p}(x,q)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{p}(x,q)}</annotation>

</semantics>

</math><img src="./7872748dc1b5ae53565bbeae6d1287802adf8288.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.25ex; height:3.009ex;" alt="{\displaystyle {\text{se}}_{p}(x,q)}" loading="lazy"> never occur for the same value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">. (See Ince's Theorem above.)

These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly.

<caption>Classification of Mathieu functions<a href="#cite_note-15">[15]</a>

</caption>

<th>Order</th>

<th>First kind</th>

<th>Second kind

</th></tr>

<tr>

<td>Integral

</td>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy">

</td>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{fe}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./dd63d2f774f9b0773d73ff81eb6161d1104f95da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.205ex; height:2.843ex;" alt="{\displaystyle {\text{fe}}_{n}(x,q)}" loading="lazy">

</td></tr>

<tr>

<td>Integral

</td>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy">

</td>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{ge}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./17f21e0b09a619f1dc7ddd8ad33ba0c263215c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.656ex; height:2.843ex;" alt="{\displaystyle {\text{ge}}_{n}(x,q)}" loading="lazy">

</td></tr>

<tr>

<td>Fractional

(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> non-integral)

</td>

<msub>

</mrow>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{p}(x,q)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{p}(x,q)}</annotation>

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</math><img src="./7872748dc1b5ae53565bbeae6d1287802adf8288.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.25ex; height:3.009ex;" alt="{\displaystyle {\text{se}}_{p}(x,q)}" loading="lazy">

</td></tr></tbody></table></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Explicit_representation_and_computation">Explicit representation and computation</h2></div>

<div class="mw-heading mw-heading3"><h3 id="First_kind">First kind</h3></div>

Mathieu functions of the first kind can be represented as <a href="Fourier_series" title="Fourier series">Fourier series</a>:<a href="#cite_note-Arscott_chapIII-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{ce}}_{2n}(x,q)&=\sum _{r=0}^{\infty }A_{2r}^{(2n)}(q)\cos(2rx)\\{\text{ce}}_{2n+1}(x,q)&=\sum _{r=0}^{\infty }A_{2r+1}^{(2n+1)}(q)\cos \left[(2r+1)x\right]\\{\text{se}}_{2n+1}(x,q)&=\sum _{r=0}^{\infty }B_{2r+1}^{(2n+1)}(q)\sin \left[(2r+1)x\right]\\{\text{se}}_{2n+2}(x,q)&=\sum _{r=0}^{\infty }B_{2r+2}^{(2n+2)}(q)\sin \left[(2r+2)x\right]\\\end{aligned}}}</annotation>

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</math><img src="./e064250cbbcd29bfa53377d1a67b649426e29433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.505ex; width:43.227ex; height:28.176ex;" alt="{\displaystyle {\begin{aligned}{\text{ce}}_{2n}(x,q)&=\sum _{r=0}^{\infty }A_{2r}^{(2n)}(q)\cos(2rx)\\{\text{ce}}_{2n+1}(x,q)&=\sum _{r=0}^{\infty }A_{2r+1}^{(2n+1)}(q)\cos \left[(2r+1)x\right]\\{\text{se}}_{2n+1}(x,q)&=\sum _{r=0}^{\infty }B_{2r+1}^{(2n+1)}(q)\sin \left[(2r+1)x\right]\\{\text{se}}_{2n+2}(x,q)&=\sum _{r=0}^{\infty }B_{2r+2}^{(2n+2)}(q)\sin \left[(2r+2)x\right]\\\end{aligned}}}" loading="lazy"></dd></dl>

The expansion coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}^{(i)}(q)}">

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</math><img src="./1d6492dabd6e55e24c0bea3d129a6612c77c7148.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.701ex; height:4.009ex;" alt="{\displaystyle A_{j}^{(i)}(q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{j}^{(i)}(q)}">

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</math><img src="./1546fe3abde4869139e6b027a7eb546657d600fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.722ex; height:4.009ex;" alt="{\displaystyle B_{j}^{(i)}(q)}" loading="lazy"> are functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> but independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. By substitution into the Mathieu equation, they can be shown to obey three-term <a href="Recurrence_relation" title="Recurrence relation">recurrence relations</a> in the lower index. For instance, for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{2n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{2n}}</annotation>

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</math><img src="./5d03004a59515e60b107b3615452127fa77c6db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.105ex; height:2.009ex;" alt="{\displaystyle {\text{ce}}_{2n}}" loading="lazy"> one finds<a href="#cite_note-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}aA_{0}-qA_{2}&=0\\(a-4)A_{2}-q(A_{4}+2A_{0})&=0\\(a-4r^{2})A_{2r}-q(A_{2r+2}+A_{2r-2})&=0,\quad r\geq 2\end{aligned}}}</annotation>

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</math><img src="./2ad0dd2983acebbce096ffa8242afcb56cb70d54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:46.201ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}aA_{0}-qA_{2}&=0\\(a-4)A_{2}-q(A_{4}+2A_{0})&=0\\(a-4r^{2})A_{2r}-q(A_{2r+2}+A_{2r-2})&=0,\quad r\geq 2\end{aligned}}}" loading="lazy"></dd></dl>

Being a second-order recurrence in the index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2r}">

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<annotation encoding="application/x-tex">{\displaystyle 2r}</annotation>

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</math><img src="./bb337de6fbd1ab48176084f9c4534b8c55847042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.211ex; height:2.176ex;" alt="{\displaystyle 2r}" loading="lazy">, one can always find two independent solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2r}}">

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</math><img src="./a9519dcaac048ebff97a32f2f2e3584db530581d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.72ex; height:2.509ex;" alt="{\displaystyle X_{2r}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{2r}}">

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</math><img src="./930ccfdaebe5c210a8ebcaee081c0d5d0efb8796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.146ex; height:2.509ex;" alt="{\displaystyle Y_{2r}}" loading="lazy"> such that the general solution can be expressed as a linear combination of the two: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2r}=c_{1}X_{2r}+c_{2}Y_{2r}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{2r}=c_{1}X_{2r}+c_{2}Y_{2r}}</annotation>

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</math><img src="./e47a06ec58a21bb103c0286f8e9e842192850cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.466ex; height:2.509ex;" alt="{\displaystyle A_{2r}=c_{1}X_{2r}+c_{2}Y_{2r}}" loading="lazy">. Moreover, in this particular case, an asymptotic analysis<a href="#cite_note-Wimp1984-17">[17]</a> shows that one possible choice of fundamental solutions has the property

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X_{2r}&=r^{-2r-1}\left(-{\frac {e^{2}q}{4}}\right)^{r}\left[1+{\mathcal {O}}(r^{-1})\right]\\Y_{2r}&=r^{2r-1}\left(-{\frac {4}{e^{2}q}}\right)^{r}\left[1+{\mathcal {O}}(r^{-1})\right]\end{aligned}}}</annotation>

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</math><img src="./4aadb90075405538f01da04fafc7a5617d4038ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:37.41ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}X_{2r}&=r^{-2r-1}\left(-{\frac {e^{2}q}{4}}\right)^{r}\left[1+{\mathcal {O}}(r^{-1})\right]\\Y_{2r}&=r^{2r-1}\left(-{\frac {4}{e^{2}q}}\right)^{r}\left[1+{\mathcal {O}}(r^{-1})\right]\end{aligned}}}" loading="lazy"></dd></dl>

In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2r}}">

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</math><img src="./a9519dcaac048ebff97a32f2f2e3584db530581d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.72ex; height:2.509ex;" alt="{\displaystyle X_{2r}}" loading="lazy"> is finite whereas <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{2r}}">

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</math><img src="./930ccfdaebe5c210a8ebcaee081c0d5d0efb8796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.146ex; height:2.509ex;" alt="{\displaystyle Y_{2r}}" loading="lazy"> diverges. Writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2r}=c_{1}X_{2r}+c_{2}Y_{2r}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{2r}=c_{1}X_{2r}+c_{2}Y_{2r}}</annotation>

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</math><img src="./e47a06ec58a21bb103c0286f8e9e842192850cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.466ex; height:2.509ex;" alt="{\displaystyle A_{2r}=c_{1}X_{2r}+c_{2}Y_{2r}}" loading="lazy">, we therefore see that in order for the Fourier series representation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{2n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{2n}}</annotation>

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</math><img src="./5d03004a59515e60b107b3615452127fa77c6db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.105ex; height:2.009ex;" alt="{\displaystyle {\text{ce}}_{2n}}" loading="lazy"> to converge, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> must be chosen such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}=0.}">

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</math><img src="./589e16d7fa2a4f2c0348bd7cf118ba9690b2de24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.969ex; height:2.509ex;" alt="{\displaystyle c_{2}=0.}" loading="lazy"> These choices of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> correspond to the characteristic numbers.

In general, however, the solution of a three-term recurrence with variable coefficients

cannot be represented in a simple manner, and hence there is no simple way to determine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> from the condition

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</math><img src="./6c225817c2d6d5cf49d17531f93b2102b24205d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.322ex; height:2.509ex;" alt="{\displaystyle c_{2}=0}" loading="lazy">. Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2r}}">

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</math><img src="./7dd4558e2efa0ca95dffdc7278ffcc1298b9578c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.539ex; height:2.509ex;" alt="{\displaystyle A_{2r}}" loading="lazy"> by numerically iterating the recurrence towards increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">. The reason is that as long as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> only approximates a characteristic number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}}">

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</math><img src="./0b30ba1b247fb8d334580cec68561e749d24aff2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{2}}" loading="lazy"> is not identically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> and the divergent solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{2r}}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{2r}}</annotation>

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</math><img src="./930ccfdaebe5c210a8ebcaee081c0d5d0efb8796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.146ex; height:2.509ex;" alt="{\displaystyle Y_{2r}}" loading="lazy"> eventually dominates for large enough <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">.

To overcome these issues, more sophisticated semi-analytical/numerical approaches are required, for instance using a <a href="Continued_fraction" title="Continued fraction">continued fraction</a> expansion,<a href="#cite_note-18">[18]</a><a href="#cite_note-Arscott_chapIII-5">[5]</a> casting the recurrence as a <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> eigenvalue problem,<a href="#cite_note-19">[19]</a> or implementing a backwards recurrence algorithm.<a href="#cite_note-Wimp1984-17">[17]</a> The complexity of the three-term recurrence relation is one of the reasons there are few simple formulas and identities involving Mathieu functions.<a href="#cite_note-Temme2015-20">[20]</a>

In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as <a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>, <a href="Maple_(software)" title="Maple (software)">Maple</a>, <a href="MATLAB" title="MATLAB">MATLAB</a>, and <a href="SciPy" title="SciPy">SciPy</a>. For small values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> and low order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, they can also be expressed perturbatively as <a href="Power_series" title="Power series">power series</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, which can be useful in physical applications.<a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading3"><h3 id="Second_kind_2">Second kind</h3></div>

There are several ways to represent Mathieu functions of the second kind.<a href="#cite_note-22">[22]</a> One representation is in terms of <a href="Bessel_function" title="Bessel function">Bessel functions</a>:<a href="#cite_note-Malits2010-23">[23]</a>

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<mtext>where </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{fe}}_{2n}(x,q)&=-{\frac {\pi \gamma _{n}}{2}}\sum _{r=0}^{\infty }(-1)^{r+n}A_{2r}^{(2n)}(-q)\ {\text{Im}}[J_{r}({\sqrt {q}}e^{ix})Y_{r}({\sqrt {q}}e^{-ix})],\quad {\text{where }}\gamma _{n}=\left\{{\begin{array}{cc}{\sqrt {2}},&{\text{ if }}n=0\\2n,&{\text{ if }}n\geq 1\end{array}}\right.\\{\text{fe}}_{2n+1}(x,q)&={\frac {\pi {\sqrt {q}}}{2}}\sum _{r=0}^{\infty }(-1)^{r+n}A_{2r+1}^{(2n+1)}(-q)\ {\text{Im}}[J_{r}({\sqrt {q}}e^{ix})Y_{r+1}({\sqrt {q}}e^{-ix})+J_{r+1}({\sqrt {q}}e^{ix})Y_{r}({\sqrt {q}}e^{-ix})]\\{\text{ge}}_{2n+1}(x,q)&=-{\frac {\pi {\sqrt {q}}}{2}}\sum _{r=0}^{\infty }(-1)^{r+n}B_{2r+1}^{(2n+1)}(-q)\ {\text{Re}}[J_{r}({\sqrt {q}}e^{ix})Y_{r+1}({\sqrt {q}}e^{-ix})-J_{r+1}({\sqrt {q}}e^{ix})Y_{r}({\sqrt {q}}e^{-ix})]\\{\text{ge}}_{2n+2}(x,q)&=-{\frac {\pi q}{4(n+1)}}\sum _{r=0}^{\infty }(-1)^{r+n}B_{2r+2}^{(2n+2)}(-q)\ {\text{Re}}[J_{r}({\sqrt {q}}e^{ix})Y_{r+2}({\sqrt {q}}e^{-ix})-J_{r+2}({\sqrt {q}}e^{ix})Y_{r}({\sqrt {q}}e^{-ix})]\end{aligned}}}</annotation>

</semantics>

</math><img src="./04e313f3d906e7123227e7d4a6982ffba2acfae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:104.019ex; height:28.509ex;" alt="{\displaystyle {\begin{aligned}{\text{fe}}_{2n}(x,q)&=-{\frac {\pi \gamma _{n}}{2}}\sum _{r=0}^{\infty }(-1)^{r+n}A_{2r}^{(2n)}(-q)\ {\text{Im}}[J_{r}({\sqrt {q}}e^{ix})Y_{r}({\sqrt {q}}e^{-ix})],\quad {\text{where }}\gamma _{n}=\left\{{\begin{array}{cc}{\sqrt {2}},&{\text{ if }}n=0\\2n,&{\text{ if }}n\geq 1\end{array}}\right.\\{\text{fe}}_{2n+1}(x,q)&={\frac {\pi {\sqrt {q}}}{2}}\sum _{r=0}^{\infty }(-1)^{r+n}A_{2r+1}^{(2n+1)}(-q)\ {\text{Im}}[J_{r}({\sqrt {q}}e^{ix})Y_{r+1}({\sqrt {q}}e^{-ix})+J_{r+1}({\sqrt {q}}e^{ix})Y_{r}({\sqrt {q}}e^{-ix})]\\{\text{ge}}_{2n+1}(x,q)&=-{\frac {\pi {\sqrt {q}}}{2}}\sum _{r=0}^{\infty }(-1)^{r+n}B_{2r+1}^{(2n+1)}(-q)\ {\text{Re}}[J_{r}({\sqrt {q}}e^{ix})Y_{r+1}({\sqrt {q}}e^{-ix})-J_{r+1}({\sqrt {q}}e^{ix})Y_{r}({\sqrt {q}}e^{-ix})]\\{\text{ge}}_{2n+2}(x,q)&=-{\frac {\pi q}{4(n+1)}}\sum _{r=0}^{\infty }(-1)^{r+n}B_{2r+2}^{(2n+2)}(-q)\ {\text{Re}}[J_{r}({\sqrt {q}}e^{ix})Y_{r+2}({\sqrt {q}}e^{-ix})-J_{r+2}({\sqrt {q}}e^{ix})Y_{r}({\sqrt {q}}e^{-ix})]\end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,q>0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n,q>0}</annotation>

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</math><img src="./8f70a28d0def9a642cc39f926e221e601b02d0be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.759ex; height:2.509ex;" alt="{\displaystyle n,q>0}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{r}(x)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle J_{r}(x)}</annotation>

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</math><img src="./9219ae5c604bbf1a402d4c8236d9949fc76c0249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.403ex; height:2.843ex;" alt="{\displaystyle J_{r}(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{r}(x)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Y_{r}(x)}</annotation>

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</math><img src="./7da97e1c9d85cb06ec12233cca66c1ed3cce7e29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.843ex;" alt="{\displaystyle Y_{r}(x)}" loading="lazy"> are Bessel functions of the first and second kind.

<div class="mw-heading mw-heading3"><h3 id="Modified_functions">Modified functions</h3></div>

A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel function product series.<a href="#cite_note-Jin1996-24">[24]</a> For large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, the form of the series must be chosen carefully to avoid subtraction errors.<a href="#cite_note-Van_BurenBoisvert2007-25">[25]</a><a href="#cite_note-Bibby2013-26">[26]</a>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

There are relatively few analytic expressions and identities involving Mathieu functions. Moreover, unlike many other <a href="Special_functions" title="Special functions">special functions</a>, the solutions of Mathieu's equation cannot in general be expressed in terms of <a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric functions</a>. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\cos(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\cos(x)}</annotation>

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</math><img src="./2fcdc1d409a9e40680508ecd59d7365d5088dd1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.188ex; height:2.843ex;" alt="{\displaystyle t=\cos(x)}" loading="lazy">:

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (1-t^{2}){\frac {d^{2}y}{dt^{2}}}-t\,{\frac {dy}{dt}}+(a+2q(1-2t^{2}))\,y=0.}</annotation>

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</math><img src="./8e39ef8e2d7394ba35d73cab4ebde7874a51cab7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:45.515ex; height:6.009ex;" alt="{\displaystyle (1-t^{2}){\frac {d^{2}y}{dt^{2}}}-t\,{\frac {dy}{dt}}+(a+2q(1-2t^{2}))\,y=0.}" loading="lazy"></dd></dl>

Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type.<a href="#cite_note-Temme2015-20">[20]</a>

<div class="mw-heading mw-heading3"><h3 id="Qualitative_behavior">Qualitative behavior</h3></div>

For small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}}</annotation>

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</math><img src="./7b3b7459759cb25e6e290ac8deac0d9402fabf51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.283ex; height:2.009ex;" alt="{\displaystyle {\text{ce}}_{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}}</annotation>

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</math><img src="./c1181d702f42d8e7ed3d39cd99b821b3dcab143d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.167ex; height:2.009ex;" alt="{\displaystyle {\text{se}}_{n}}" loading="lazy"> behave similarly to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos nx}">

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<annotation encoding="application/x-tex">{\displaystyle \cos nx}</annotation>

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</math><img src="./a08b9fef43f2ff87e3d5e980ebe2f635fbee0e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.223ex; height:1.676ex;" alt="{\displaystyle \cos nx}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin nx}">

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<annotation encoding="application/x-tex">{\displaystyle \sin nx}</annotation>

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</math><img src="./fc4cf3dbda60f1aa597014807d61ccfee5be156e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.967ex; height:2.176ex;" alt="{\displaystyle \sin nx}" loading="lazy">. For arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{m}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{m}(x,q)}</annotation>

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</math><img src="./7029361b3ab58cbb41dc2b3bc06987f0c26121e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.982ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{m}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{m+1}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{m+1}(x,q)}</annotation>

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</math><img src="./756c930503399f332e46b6ba0479644ea389a573.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.967ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{m+1}(x,q)}" loading="lazy"> have exactly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> <a href="Simple_zero" class="mw-redirect" title="Simple zero">simple zeros</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x<\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 0<x<\pi }</annotation>

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</math><img src="./a5072e331d80251147f44dc1bca9faa5d19c96a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.021ex; height:2.176ex;" alt="{\displaystyle 0<x<\pi }" loading="lazy">, and as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\rightarrow \infty }">

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<annotation encoding="application/x-tex">{\displaystyle q\rightarrow \infty }</annotation>

</semantics>

</math><img src="./4ca99a8721a2fcbdf3dc549eb4742b040948cfd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.007ex; height:2.176ex;" alt="{\displaystyle q\rightarrow \infty }" loading="lazy"> the zeros cluster about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pi /2}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle x=\pi /2}</annotation>

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</math><img src="./d97d5442f1430539346bc2ad4db5114c16882dad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.085ex; height:2.843ex;" alt="{\displaystyle x=\pi /2}" loading="lazy">.<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q>0}">

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<annotation encoding="application/x-tex">{\displaystyle q>0}</annotation>

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</math><img src="./482e0a33d9e8fd6307b5f68a5182c2d0d14efc9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q>0}" loading="lazy"> and as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\rightarrow \infty }">

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<annotation encoding="application/x-tex">{\displaystyle x\rightarrow \infty }</annotation>

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</math><img src="./02579e74e2ef1ca0befceba816b311fe5bfd6844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\rightarrow \infty }" loading="lazy"> the modified Mathieu functions tend to behave as damped periodic functions.

In the following, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> factors from the Fourier expansions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}}</annotation>

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</math><img src="./7b3b7459759cb25e6e290ac8deac0d9402fabf51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.283ex; height:2.009ex;" alt="{\displaystyle {\text{ce}}_{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}}</annotation>

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</math><img src="./c1181d702f42d8e7ed3d39cd99b821b3dcab143d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.167ex; height:2.009ex;" alt="{\displaystyle {\text{se}}_{n}}" loading="lazy"> may be referenced (see <a href="#Explicit_representation_and_computation">Explicit representation and computation</a>). They depend on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> but are independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Reflections_and_translations">Reflections and translations</h3></div>

Due to their parity and periodicity, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}}</annotation>

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</math><img src="./7b3b7459759cb25e6e290ac8deac0d9402fabf51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.283ex; height:2.009ex;" alt="{\displaystyle {\text{ce}}_{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}}</annotation>

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</math><img src="./c1181d702f42d8e7ed3d39cd99b821b3dcab143d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.167ex; height:2.009ex;" alt="{\displaystyle {\text{se}}_{n}}" loading="lazy"> have simple properties under reflections and translations by multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">:<a href="#cite_note-McLachlan_chapII-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\text{ce}}_{n}(x+\pi )=(-1)^{n}{\text{ce}}_{n}(x)\\&{\text{se}}_{n}(x+\pi )=(-1)^{n}{\text{se}}_{n}(x)\\&{\text{ce}}_{n}(x+\pi /2)=(-1)^{n}{\text{ce}}_{n}(-x+\pi /2)\\&{\text{se}}_{n+1}(x+\pi /2)=(-1)^{n}{\text{se}}_{n+1}(-x+\pi /2)\end{aligned}}}</annotation>

</semantics>

</math><img src="./0fd1c02dfb4e8976905667c32eb0b9660776ffa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:41.464ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}&{\text{ce}}_{n}(x+\pi )=(-1)^{n}{\text{ce}}_{n}(x)\\&{\text{se}}_{n}(x+\pi )=(-1)^{n}{\text{se}}_{n}(x)\\&{\text{ce}}_{n}(x+\pi /2)=(-1)^{n}{\text{ce}}_{n}(-x+\pi /2)\\&{\text{se}}_{n+1}(x+\pi /2)=(-1)^{n}{\text{se}}_{n+1}(-x+\pi /2)\end{aligned}}}" loading="lazy"></dd></dl>

One can also write functions with negative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> in terms of those with positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">:<a href="#cite_note-Arscott_chapIII-5">[5]</a><a href="#cite_note-29">[29]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\text{ce}}_{2n+1}(x,-q)=(-1)^{n}{\text{se}}_{2n+1}(-x+\pi /2,q)\\&{\text{ce}}_{2n+2}(x,-q)=(-1)^{n}{\text{ce}}_{2n+2}(-x+\pi /2,q)\\&{\text{se}}_{2n+1}(x,-q)=(-1)^{n}{\text{ce}}_{2n+1}(-x+\pi /2,q)\\&{\text{se}}_{2n+2}(x,-q)=(-1)^{n}{\text{se}}_{2n+2}(-x+\pi /2,q)\end{aligned}}}</annotation>

</semantics>

</math><img src="./57819f3b1506723be98936a1b735c3f968b7701b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:42.858ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}&{\text{ce}}_{2n+1}(x,-q)=(-1)^{n}{\text{se}}_{2n+1}(-x+\pi /2,q)\\&{\text{ce}}_{2n+2}(x,-q)=(-1)^{n}{\text{ce}}_{2n+2}(-x+\pi /2,q)\\&{\text{se}}_{2n+1}(x,-q)=(-1)^{n}{\text{ce}}_{2n+1}(-x+\pi /2,q)\\&{\text{se}}_{2n+2}(x,-q)=(-1)^{n}{\text{se}}_{2n+2}(-x+\pi /2,q)\end{aligned}}}" loading="lazy"></dd></dl>

Moreover,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&a_{2n+1}(q)=b_{2n+1}(-q)\\&b_{2n+2}(q)=b_{2n+2}(-q)\end{aligned}}}</annotation>

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</math><img src="./7bab018ed03c66e2bedd146e9800cfa47d37101f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.925ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&a_{2n+1}(q)=b_{2n+1}(-q)\\&b_{2n+2}(q)=b_{2n+2}(-q)\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Orthogonality_and_completeness">Orthogonality and completeness</h3></div>

Like their trigonometric counterparts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos nx}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \cos nx}</annotation>

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</math><img src="./a08b9fef43f2ff87e3d5e980ebe2f635fbee0e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.223ex; height:1.676ex;" alt="{\displaystyle \cos nx}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin nx}">

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<annotation encoding="application/x-tex">{\displaystyle \sin nx}</annotation>

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</math><img src="./fc4cf3dbda60f1aa597014807d61ccfee5be156e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.967ex; height:2.176ex;" alt="{\displaystyle \sin nx}" loading="lazy">, the periodic Mathieu functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}(x,q)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

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</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

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</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy"> satisfy orthogonality relations

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\int _{0}^{2\pi }{\text{ce}}_{n}{\text{ce}}_{m}\,dx=\int _{0}^{2\pi }{\text{se}}_{n}{\text{se}}_{m}\,dx=\delta _{nm}\pi \\&\int _{0}^{2\pi }{\text{ce}}_{n}{\text{se}}_{m}\,dx=0\end{aligned}}}</annotation>

</semantics>

</math><img src="./411a9b755f9bf131d746a2760cabf230618941e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:41.735ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}&\int _{0}^{2\pi }{\text{ce}}_{n}{\text{ce}}_{m}\,dx=\int _{0}^{2\pi }{\text{se}}_{n}{\text{se}}_{m}\,dx=\delta _{nm}\pi \\&\int _{0}^{2\pi }{\text{ce}}_{n}{\text{se}}_{m}\,dx=0\end{aligned}}}" loading="lazy"></dd></dl>

Moreover, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> fixed and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> treated as the eigenvalue, the Mathieu equation is of <a href="Sturm%E2%80%93Liouville" class="mw-redirect" title="Sturm–Liouville">Sturm–Liouville</a> form. This implies that the eigenfunctions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}(x,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

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</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

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</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy"> form a complete set, i.e. any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">- or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">-periodic function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> can be expanded as a series in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ce}}_{n}(x,q)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{ce}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./b31edfa46352bd059d921609901b835167d940f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.526ex; height:2.843ex;" alt="{\displaystyle {\text{ce}}_{n}(x,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{se}}_{n}(x,q)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{se}}_{n}(x,q)}</annotation>

</semantics>

</math><img src="./d42634cf37181f8558e06d21801a89e516ac411d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.41ex; height:2.843ex;" alt="{\displaystyle {\text{se}}_{n}(x,q)}" loading="lazy">.<a href="#cite_note-Gutiérrez-Vega2015-4">[4]</a>

<div class="mw-heading mw-heading3"><h3 id="Integral_identities">Integral identities</h3></div>

Solutions of Mathieu's equation satisfy a class of integral identities with respect to <a href="Integral_kernel" class="mw-redirect" title="Integral kernel">kernels</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (x,x')}">

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<annotation encoding="application/x-tex">{\displaystyle \chi (x,x')}</annotation>

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</math><img src="./435e0133f9353c0c894460f67536acaeec78c8be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.642ex; height:3.009ex;" alt="{\displaystyle \chi (x,x')}" loading="lazy"> that are solutions of

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\chi }{\partial x^{2}}}-{\frac {\partial ^{2}\chi }{\partial x'^{2}}}=2q\left(\cos 2x-\cos 2x'\right)\chi }</annotation>

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</math><img src="./38b52a376f5230d0c49e81a04b1d2720c4de6046.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.395ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\chi }{\partial x^{2}}}-{\frac {\partial ^{2}\chi }{\partial x'^{2}}}=2q\left(\cos 2x-\cos 2x'\right)\chi }" loading="lazy"></dd></dl>

More precisely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation>

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</math><img src="./546b660b2f3cfb5f34be7b3ed8371d54f5c74227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}" loading="lazy"> solves Mathieu's equation with given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, then the integral

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)\equiv \int _{C}\chi (x,x')\phi (x')dx'}</annotation>

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</math><img src="./442f885765bbc3072c7aaee1dfc2c0ee9fbcf22b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.993ex; height:5.676ex;" alt="{\displaystyle \psi (x)\equiv \int _{C}\chi (x,x')\phi (x')dx'}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is a path in the <a href="Complex_plane" title="Complex plane">complex plane</a>, also solves Mathieu's equation with the same <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, provided the following conditions are met:<a href="#cite_note-30">[30]</a>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \chi (x,x')}</annotation>

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</math><img src="./435e0133f9353c0c894460f67536acaeec78c8be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.642ex; height:3.009ex;" alt="{\displaystyle \chi (x,x')}" loading="lazy"> solves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\chi }{\partial x^{2}}}-{\frac {\partial ^{2}\chi }{\partial x'^{2}}}=2q\left(\cos 2x-\cos 2x'\right)\chi }">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\chi }{\partial x^{2}}}-{\frac {\partial ^{2}\chi }{\partial x'^{2}}}=2q\left(\cos 2x-\cos 2x'\right)\chi }</annotation>

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</math><img src="./38b52a376f5230d0c49e81a04b1d2720c4de6046.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.395ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\chi }{\partial x^{2}}}-{\frac {\partial ^{2}\chi }{\partial x'^{2}}}=2q\left(\cos 2x-\cos 2x'\right)\chi }" loading="lazy"></li>

<li>In the regions under consideration, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation>

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</math><img src="./a596a1fb4130a47f6b88c66150497338bd6cbccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}" loading="lazy"> exists and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (x,x')}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \chi (x,x')}</annotation>

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</math><img src="./435e0133f9353c0c894460f67536acaeec78c8be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.642ex; height:3.009ex;" alt="{\displaystyle \chi (x,x')}" loading="lazy"> is <a href="Analytic_function" title="Analytic function">analytic</a></li>

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<annotation encoding="application/x-tex">{\displaystyle \left(\phi {\frac {\partial \chi }{\partial x'}}-{\frac {\partial \phi }{\partial x'}}\chi \right)}</annotation>

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</math><img src="./fd8518242f849fe92d11a479811bd99fa442210c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.439ex; height:6.176ex;" alt="{\displaystyle \left(\phi {\frac {\partial \chi }{\partial x'}}-{\frac {\partial \phi }{\partial x'}}\chi \right)}" loading="lazy"> has the same value at the endpoints of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></li></ul>

Using an appropriate change of variables, the equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> can be transformed into the <a href="Wave_equation" title="Wave equation">wave equation</a> and solved. For instance, one solution is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (x,x')=\sinh(2q^{1/2}\sin x\sin x')}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \chi (x,x')=\sinh(2q^{1/2}\sin x\sin x')}</annotation>

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</math><img src="./2a15afd7f441eb671eba1664077f5b59d4fa89eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.242ex; height:3.343ex;" alt="{\displaystyle \chi (x,x')=\sinh(2q^{1/2}\sin x\sin x')}" loading="lazy">. Examples of identities obtained in this way are<a href="#cite_note-31">[31]</a>

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\text{se}}_{2n+1}(x,q)&={\frac {{\text{se}}'_{2n+1}(0,q)}{\pi q^{1/2}B_{1}^{(2n+1)}}}\int _{0}^{\pi }\sinh(2q^{1/2}\sin x\sin x'){\text{se}}_{2n+1}(x',q)dx'\qquad (q>0)\\{\text{Ce}}_{2n}(x,q)&={\frac {{\text{ce}}_{2n}(\pi /2,q)}{\pi A_{0}^{(2n)}}}\int _{0}^{\pi }\cos(2q^{1/2}\cosh x\cos x'){\text{ce}}_{2n}(x',q)dx'\qquad \ \ \ (q>0)\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{se}}_{2n+1}(x,q)&={\frac {{\text{se}}'_{2n+1}(0,q)}{\pi q^{1/2}B_{1}^{(2n+1)}}}\int _{0}^{\pi }\sinh(2q^{1/2}\sin x\sin x'){\text{se}}_{2n+1}(x',q)dx'\qquad (q>0)\\{\text{Ce}}_{2n}(x,q)&={\frac {{\text{ce}}_{2n}(\pi /2,q)}{\pi A_{0}^{(2n)}}}\int _{0}^{\pi }\cos(2q^{1/2}\cosh x\cos x'){\text{ce}}_{2n}(x',q)dx'\qquad \ \ \ (q>0)\end{aligned}}}</annotation>

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</math><img src="./4ece4536e4776dc33aad9839a20ad021142da2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:81.258ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}{\text{se}}_{2n+1}(x,q)&={\frac {{\text{se}}'_{2n+1}(0,q)}{\pi q^{1/2}B_{1}^{(2n+1)}}}\int _{0}^{\pi }\sinh(2q^{1/2}\sin x\sin x'){\text{se}}_{2n+1}(x',q)dx'\qquad (q>0)\\{\text{Ce}}_{2n}(x,q)&={\frac {{\text{ce}}_{2n}(\pi /2,q)}{\pi A_{0}^{(2n)}}}\int _{0}^{\pi }\cos(2q^{1/2}\cosh x\cos x'){\text{ce}}_{2n}(x',q)dx'\qquad \ \ \ (q>0)\end{aligned}}}" loading="lazy"></dd></dl>

Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions.<a href="#cite_note-32">[32]</a>

There also exist integral relations between functions of the first and second kind, for instance:<a href="#cite_note-Malits2010-23">[23]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\text{fe}}_{2n}(x,q)=2n\int _{0}^{x}{\text{ce}}_{2n}(\tau ,-q)\ J_{0}\left({\sqrt {2q(\cos 2x-\cos 2\tau )}}\right)d\tau ,\qquad n\geq 1}</annotation>

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</math><img src="./b184d9656cb6504db81f76e14aa8b9c893596bd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:71.407ex; height:5.843ex;" alt="{\displaystyle {\text{fe}}_{2n}(x,q)=2n\int _{0}^{x}{\text{ce}}_{2n}(\tau ,-q)\ J_{0}\left({\sqrt {2q(\cos 2x-\cos 2\tau )}}\right)d\tau ,\qquad n\geq 1}" loading="lazy"></dd></dl>

valid for any complex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Asymptotic_expansions">Asymptotic expansions</h3></div>

The following asymptotic expansions hold for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q>0}">

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<annotation encoding="application/x-tex">{\displaystyle q>0}</annotation>

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</math><img src="./482e0a33d9e8fd6307b5f68a5182c2d0d14efc9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q>0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Im}}(x)=0}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Im}}(x)=0}</annotation>

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</math><img src="./73109b09578c1509f2e0588e20e869f7932b15d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.175ex; height:2.843ex;" alt="{\displaystyle {\text{Im}}(x)=0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Re}}(x)\rightarrow \infty }">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Re}}(x)\rightarrow \infty }</annotation>

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</math><img src="./da4cf3567e521e260ceb5391c0af4002574df637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.82ex; height:2.843ex;" alt="{\displaystyle {\text{Re}}(x)\rightarrow \infty }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2q^{1/2}\cosh x\simeq q^{1/2}e^{x}}">

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<annotation encoding="application/x-tex">{\displaystyle 2q^{1/2}\cosh x\simeq q^{1/2}e^{x}}</annotation>

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</math><img src="./526bdfa4f9c24f934438fb540806226ac9f5d0d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.58ex; height:3.176ex;" alt="{\displaystyle 2q^{1/2}\cosh x\simeq q^{1/2}e^{x}}" loading="lazy">:<a href="#cite_note-McLachlan_chapXI-33">[33]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{Ce}}_{2n}(x,q)&\sim \left({\frac {2}{\pi q^{1/2}}}\right)^{1/2}{\frac {{\text{ce}}_{2n}(0,q){\text{ce}}_{2n}(\pi /2,q)}{A_{0}^{(2n)}}}\cdot e^{-x/2}\sin \left(q^{1/2}e^{x}+{\frac {\pi }{4}}\right)\\{\text{Ce}}_{2n+1}(x,q)&\sim \left({\frac {2}{\pi q^{3/2}}}\right)^{1/2}{\frac {{\text{ce}}_{2n+1}(0,q){\text{ce}}'_{2n+1}(\pi /2,q)}{A_{1}^{(2n+1)}}}\cdot e^{-x/2}\cos \left(q^{1/2}e^{x}+{\frac {\pi }{4}}\right)\\{\text{Se}}_{2n+1}(x,q)&\sim -\left({\frac {2}{\pi q^{3/2}}}\right)^{1/2}{\frac {{\text{se}}'_{2n+1}(0,q){\text{se}}_{2n+1}(\pi /2,q)}{B_{1}^{(2n+1)}}}\cdot e^{-x/2}\cos \left(q^{1/2}e^{x}+{\frac {\pi }{4}}\right)\\{\text{Se}}_{2n+2}(x,q)&\sim \left({\frac {2}{\pi q^{5/2}}}\right)^{1/2}{\frac {{\text{se}}'_{2n+2}(0,q){\text{se}}'_{2n+2}(\pi /2,q)}{B_{2}^{(2n+2)}}}\cdot e^{-x/2}\sin \left(q^{1/2}e^{x}+{\frac {\pi }{4}}\right)\end{aligned}}}</annotation>

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</math><img src="./f5d536512cff9bdeaea3014cbdee47d5afb27005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.831ex; margin-bottom: -0.174ex; width:79.651ex; height:31.176ex;" alt="{\displaystyle {\begin{aligned}{\text{Ce}}_{2n}(x,q)&\sim \left({\frac {2}{\pi q^{1/2}}}\right)^{1/2}{\frac {{\text{ce}}_{2n}(0,q){\text{ce}}_{2n}(\pi /2,q)}{A_{0}^{(2n)}}}\cdot e^{-x/2}\sin \left(q^{1/2}e^{x}+{\frac {\pi }{4}}\right)\\{\text{Ce}}_{2n+1}(x,q)&\sim \left({\frac {2}{\pi q^{3/2}}}\right)^{1/2}{\frac {{\text{ce}}_{2n+1}(0,q){\text{ce}}'_{2n+1}(\pi /2,q)}{A_{1}^{(2n+1)}}}\cdot e^{-x/2}\cos \left(q^{1/2}e^{x}+{\frac {\pi }{4}}\right)\\{\text{Se}}_{2n+1}(x,q)&\sim -\left({\frac {2}{\pi q^{3/2}}}\right)^{1/2}{\frac {{\text{se}}'_{2n+1}(0,q){\text{se}}_{2n+1}(\pi /2,q)}{B_{1}^{(2n+1)}}}\cdot e^{-x/2}\cos \left(q^{1/2}e^{x}+{\frac {\pi }{4}}\right)\\{\text{Se}}_{2n+2}(x,q)&\sim \left({\frac {2}{\pi q^{5/2}}}\right)^{1/2}{\frac {{\text{se}}'_{2n+2}(0,q){\text{se}}'_{2n+2}(\pi /2,q)}{B_{2}^{(2n+2)}}}\cdot e^{-x/2}\sin \left(q^{1/2}e^{x}+{\frac {\pi }{4}}\right)\end{aligned}}}" loading="lazy"></dd></dl>

Thus, the modified Mathieu functions decay exponentially for large real argument. Similar asymptotic expansions can be written down for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Fe}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Fe}}_{n}}</annotation>

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</math><img src="./3d3d2621c329d4b8d02874884e604c8919c74e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.769ex; height:2.509ex;" alt="{\displaystyle {\text{Fe}}_{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ge}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Ge}}_{n}}</annotation>

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</math><img src="./f6698091ef62308c4cc20021931bb3320bb0ad49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.075ex; height:2.509ex;" alt="{\displaystyle {\text{Ge}}_{n}}" loading="lazy">; these also decay exponentially for large real argument.

For the even and odd periodic Mathieu functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ce,se}">

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</math><img src="./28bab1d2fcd5075adc2bb406025c081a9d6b478a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.298ex; height:2.009ex;" alt="{\displaystyle ce,se}" loading="lazy"> and the associated characteristic numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> one can also derive asymptotic expansions for large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">.<a href="#cite_note-34">[34]</a> For the characteristic numbers in particular, one has with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> approximately an odd integer, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\approx N_{0}=2n+1,n=1,2,3,...,}">

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<annotation encoding="application/x-tex">{\displaystyle N\approx N_{0}=2n+1,n=1,2,3,...,}</annotation>

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</math><img src="./b843afa24a7c009e47b12dc253393ca3674ded69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.606ex; height:2.509ex;" alt="{\displaystyle N\approx N_{0}=2n+1,n=1,2,3,...,}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a(N)={}&-2q+2q^{1/2}N-{\frac {1}{2^{3}}}(N^{2}+1)-{\frac {1}{2^{7}q^{1/2}}}N(N^{2}+3)-{\frac {1}{2^{12}q}}(5N^{4}+34N^{2}+9)\\&-{\frac {1}{2^{17}q^{3/2}}}N(33N^{4}+410N^{2}+405)-{\frac {1}{2^{20}q^{2}}}(63N^{6}+1260N^{4}+2943N^{2}+41807)+{\mathcal {O}}(q^{-5/2})\end{aligned}}}</annotation>

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</math><img src="./c00274acff3450f2ca0731727aa68222fcabdbe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:102.357ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}a(N)={}&-2q+2q^{1/2}N-{\frac {1}{2^{3}}}(N^{2}+1)-{\frac {1}{2^{7}q^{1/2}}}N(N^{2}+3)-{\frac {1}{2^{12}q}}(5N^{4}+34N^{2}+9)\\&-{\frac {1}{2^{17}q^{3/2}}}N(33N^{4}+410N^{2}+405)-{\frac {1}{2^{20}q^{2}}}(63N^{6}+1260N^{4}+2943N^{2}+41807)+{\mathcal {O}}(q^{-5/2})\end{aligned}}}" loading="lazy"></dd></dl>

Observe the symmetry here in replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{1/2}}">

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</math><img src="./2b0d4537fc246ba00a07b0b65fd59999272ee698.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.778ex; height:3.176ex;" alt="{\displaystyle q^{1/2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -q^{1/2}}">

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<annotation encoding="application/x-tex">{\displaystyle -q^{1/2}}</annotation>

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</math><img src="./f2ca52867f98f4adb4d2c776bcd8c0dcd749d15e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.586ex; height:3.176ex;" alt="{\displaystyle -q^{1/2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -N}">

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<annotation encoding="application/x-tex">{\displaystyle -N}</annotation>

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</math><img src="./8a714f7426fd09d79faf269548a77d4a7047919f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.872ex; height:2.343ex;" alt="{\displaystyle -N}" loading="lazy">, which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |q|^{-7/2}}">

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<annotation encoding="application/x-tex">{\displaystyle |q|^{-7/2}}</annotation>

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</math><img src="./0b45c8d331b6db68387a827e71e49b57ee115614.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.34ex; height:3.509ex;" alt="{\displaystyle |q|^{-7/2}}" loading="lazy">.<a href="#cite_note-Dingle_and_Müller_1962-35">[35]</a> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is only approximately an odd integer because in the limit of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle q\to \infty }</annotation>

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</math><img src="./953a2f08f9d9ab42fcec5c6afeea8cb551e42c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.007ex; height:2.176ex;" alt="{\displaystyle q\to \infty }" loading="lazy"> all minimum segments of the periodic potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 2x}">

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<annotation encoding="application/x-tex">{\displaystyle \cos 2x}</annotation>

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</math><img src="./44ac7248b57d247ff16d5bc299ddaf1760f9a600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.99ex; height:2.176ex;" alt="{\displaystyle \cos 2x}" loading="lazy"> become effectively independent harmonic oscillators (hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle N_{0}}</annotation>

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</math><img src="./5b6328fbe0cded37216c90735c89ee188be26a30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}" loading="lazy"> an odd integer). By decreasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\to a_{\mp }}">

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</math><img src="./a69de1fe0ba127c14e1919f04fc461a2e8bb3a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.585ex; height:2.343ex;" alt="{\displaystyle a\to a_{\mp }}" loading="lazy"> (in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions<a href="#cite_note-Dingle_and_Müller_1962-35">[35]</a> (in quantum mechanics this provides the splitting of the eigenvalues into energy bands).<a href="#cite_note-Müller-Kirsten_2012-36">[36]</a> The boundary conditions are:

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {dce_{N_{0}-1}}{dx}}\right)_{\pi /2}=0,\;\;ce_{N_{0}}(\pi /2)=0,\;\;\left({\frac {dse_{N_{0}}}{dx}}\right)_{\pi /2}=0,\;\;se_{N_{0}+1}(\pi /2)=0.}</annotation>

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</math><img src="./3f66b6d4ea2eeceddcd0fa615feb4042b8ea4337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:74.442ex; height:6.843ex;" alt="{\displaystyle \left({\frac {dce_{N_{0}-1}}{dx}}\right)_{\pi /2}=0,\;\;ce_{N_{0}}(\pi /2)=0,\;\;\left({\frac {dse_{N_{0}}}{dx}}\right)_{\pi /2}=0,\;\;se_{N_{0}+1}(\pi /2)=0.}" loading="lazy"></dd></dl>

Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> one obtains

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<annotation encoding="application/x-tex">{\displaystyle N-N_{0}=\mp 2\left({\frac {2}{\pi }}\right)^{1/2}{\frac {(16q^{1/2})^{N_{0}/2}e^{-4q^{1/2}}}{[{\frac {1}{2}}(N_{0}-1)]!}}\left[1-{\frac {3(N_{0}^{2}+1)}{2^{6}q^{1/2}}}+{\frac {1}{2^{13}q}}(9N_{0}^{4}-40N_{0}^{3}+18N_{0}^{2}-136N_{0}+9)+\dots \right].}</annotation>

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</math><img src="./dc8467b449a54ac5188886ce5133fb16245ca67b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:109.858ex; height:7.843ex;" alt="{\displaystyle N-N_{0}=\mp 2\left({\frac {2}{\pi }}\right)^{1/2}{\frac {(16q^{1/2})^{N_{0}/2}e^{-4q^{1/2}}}{[{\frac {1}{2}}(N_{0}-1)]!}}\left[1-{\frac {3(N_{0}^{2}+1)}{2^{6}q^{1/2}}}+{\frac {1}{2^{13}q}}(9N_{0}^{4}-40N_{0}^{3}+18N_{0}^{2}-136N_{0}+9)+\dots \right].}" loading="lazy"></dd></dl>

The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.

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<annotation encoding="application/x-tex">{\displaystyle a(N)=a(N_{0})+(N-N_{0})\left({\frac {\partial a}{\partial N}}\right)_{N_{0}}+\cdots .}</annotation>

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</math><img src="./0bf08e12f86df331e5c0d5b113a87495b62653fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:43.255ex; height:6.509ex;" alt="{\displaystyle a(N)=a(N_{0})+(N-N_{0})\left({\frac {\partial a}{\partial N}}\right)_{N_{0}}+\cdots .}" loading="lazy"></dd></dl>

Insertion of the appropriate expressions above yields the result

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a(N)\to a_{\mp }(N_{0})={}&-2q+2q^{1/2}N_{0}-{\frac {1}{2^{3}}}(N_{0}^{2}+1)-{\frac {1}{2^{7}q^{1/2}}}N_{0}(N_{0}^{2}+3)-{\frac {1}{2^{12}q}}(5N_{0}^{4}+34N_{0}^{2}+9)-\cdots \\&\mp {\frac {(16q^{1/2})^{N_{0}/2+1}e^{-4q^{1/2}}}{(8\pi )^{1/2}[{\frac {1}{2}}(N_{0}-1)]!}}{\bigg [}1-{\frac {N_{0}}{2^{6}q^{1/2}}}(3N_{0}^{2}+8N_{0}+3)+\cdots {\bigg ]}.\end{aligned}}}</annotation>

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</math><img src="./1f854ae0bf894b49eceecdd00154abfbb07f05d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.277ex; margin-bottom: -0.228ex; width:104.116ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}a(N)\to a_{\mp }(N_{0})={}&-2q+2q^{1/2}N_{0}-{\frac {1}{2^{3}}}(N_{0}^{2}+1)-{\frac {1}{2^{7}q^{1/2}}}N_{0}(N_{0}^{2}+3)-{\frac {1}{2^{12}q}}(5N_{0}^{4}+34N_{0}^{2}+9)-\cdots \\&\mp {\frac {(16q^{1/2})^{N_{0}/2+1}e^{-4q^{1/2}}}{(8\pi )^{1/2}[{\frac {1}{2}}(N_{0}-1)]!}}{\bigg [}1-{\frac {N_{0}}{2^{6}q^{1/2}}}(3N_{0}^{2}+8N_{0}+3)+\cdots {\bigg ]}.\end{aligned}}}" loading="lazy"></dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{0}=1,3,5,\dots }">

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<annotation encoding="application/x-tex">{\displaystyle N_{0}=1,3,5,\dots }</annotation>

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</math><img src="./d201a2f1b6da61a635da43d50d48217329205062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.331ex; height:2.509ex;" alt="{\displaystyle N_{0}=1,3,5,\dots }" loading="lazy"> these are the eigenvalues associated with the even Mathieu eigenfunctions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ce_{N_{0}}}">

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<annotation encoding="application/x-tex">{\displaystyle ce_{N_{0}}}</annotation>

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</math><img src="./59032a0313f7478d40c95477368cc05f5bbb92fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.474ex; height:2.343ex;" alt="{\displaystyle ce_{N_{0}}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ce_{N_{0}-1}}">

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<annotation encoding="application/x-tex">{\displaystyle ce_{N_{0}-1}}</annotation>

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</math><img src="./93ee73557778a0f5eb299161fc89a0fa6d2da19d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.574ex; height:2.343ex;" alt="{\displaystyle ce_{N_{0}-1}}" loading="lazy"> (i.e. with upper, minus sign) and odd Mathieu eigenfunctions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle se_{N_{0}+1}}">

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<annotation encoding="application/x-tex">{\displaystyle se_{N_{0}+1}}</annotation>

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</math><img src="./56741f6a10d988a0c71e832e3c637063bb7de337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.658ex; height:2.343ex;" alt="{\displaystyle se_{N_{0}+1}}" loading="lazy"> or

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<annotation encoding="application/x-tex">{\displaystyle se_{N_{0}}}</annotation>

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</math><img src="./d684d329c0c97bc3c175d6d42fe787a46a0fed0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.557ex; height:2.343ex;" alt="{\displaystyle se_{N_{0}}}" loading="lazy"> (i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in <a href="#cite_note-Dingle_and_Müller_1962-35">[35]</a> or.<a href="#cite_note-Müller-Kirsten_2012-36">[36]</a>

Similar asymptotic expansions can be obtained for the solutions of other periodic differential equations, as for <a href="Lam%C3%A9_function" title="Lamé function">Lamé functions</a> and prolate and oblate <a href="Spheroidal_wave_function" title="Spheroidal wave function">spheroidal wave functions</a>.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a>. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time. Examples within both categories are discussed below.

<div class="mw-heading mw-heading3"><h3 id="Partial_differential_equations">Partial differential equations</h3></div>

Mathieu functions arise when <a href="Separation_of_variables" title="Separation of variables">separation of variables</a> in elliptic coordinates is applied to 1) the <a href="Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a> in 3 dimensions, and 2) the <a href="Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a> in either 2 or 3 dimensions. Since the Helmholtz equation is a prototypical equation for modeling the spatial variation of classical waves, Mathieu functions can be used to describe a variety of wave phenomena. For instance, in <a href="Computational_electromagnetics" title="Computational electromagnetics">computational electromagnetics</a> they can be used to analyze the <a href="Scattering" title="Scattering">scattering</a> of <a href="Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic waves</a> off elliptic cylinders, and wave propagation in elliptic <a href="Waveguide" title="Waveguide">waveguides</a>.<a href="#cite_note-37">[37]</a> In <a href="General_relativity" title="General relativity">general relativity</a>, an exact plane wave solution to the <a href="Einstein_field_equation" class="mw-redirect" title="Einstein field equation">Einstein field equation</a> can be given in terms of Mathieu functions.

More recently, Mathieu functions have been used to solve a special case of the <a href="Smoluchowski_equation" class="mw-redirect" title="Smoluchowski equation">Smoluchowski equation</a>, describing the steady-state statistics of <a href="Self-propelled_particle" class="mw-redirect" title="Self-propelled particle">self-propelled particles</a>.<a href="#cite_note-38">[38]</a>

The remainder of this section details the analysis for the two-dimensional Helmholtz equation.<a href="#cite_note-39">[39]</a> In rectangular coordinates, the Helmholtz equation is

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)\psi +k^{2}\psi =0,}</annotation>

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</math><img src="./207ba5670f964ddc1a337a024f849d9759ab8188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.596ex; height:6.343ex;" alt="{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)\psi +k^{2}\psi =0,}" loading="lazy"></dd></dl>

<a href="Elliptic_coordinates" class="mw-redirect" title="Elliptic coordinates">Elliptic coordinates</a> are defined by

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=c\cosh \mu \cos \nu \\y&=c\sinh \mu \sin \nu \end{aligned}}}</annotation>

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</math><img src="./825bdf0f356f36c184ae609582d90ea5cc70470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.883ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}x&=c\cosh \mu \cos \nu \\y&=c\sinh \mu \sin \nu \end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \mu <\infty }">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq \mu <\infty }</annotation>

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</math><img src="./d17e2bfb5de75e4bd385862fb4cfe84f1fb89b14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.085ex; height:2.676ex;" alt="{\displaystyle 0\leq \mu <\infty }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \nu <2\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq \nu <2\pi }</annotation>

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</math><img src="./743f81d594d02bb290b4d2915d622d51288125f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.086ex; height:2.343ex;" alt="{\displaystyle 0\leq \nu <2\pi }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is a positive constant. The Helmholtz equation in these coordinates is

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{c^{2}(\sinh ^{2}\mu +\sin ^{2}\nu )}}\left({\frac {\partial ^{2}}{\partial \mu ^{2}}}+{\frac {\partial ^{2}}{\partial \nu ^{2}}}\right)\psi +k^{2}\psi =0}</annotation>

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</math><img src="./6d658144b2169f99100d19ab56660e42c9e0dfe3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.572ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{c^{2}(\sinh ^{2}\mu +\sin ^{2}\nu )}}\left({\frac {\partial ^{2}}{\partial \mu ^{2}}}+{\frac {\partial ^{2}}{\partial \nu ^{2}}}\right)\psi +k^{2}\psi =0}" loading="lazy"></dd></dl>

The constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> curves are <a href="Confocal_conic_sections" title="Confocal conic sections">confocal ellipses</a> with focal length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries. Separation of variables via <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mu ,\nu )=F(\mu )G(\nu )}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mu ,\nu )=F(\mu )G(\nu )}</annotation>

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</math><img src="./acdccee9fb7be19e79358a91682bde1dae44966f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.909ex; height:2.843ex;" alt="{\displaystyle \psi (\mu ,\nu )=F(\mu )G(\nu )}" loading="lazy"> yields the Mathieu equations

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {d^{2}F}{d\mu ^{2}}}-\left(a-{\frac {c^{2}k^{2}}{2}}\cosh 2\mu \right)F=0\\&{\frac {d^{2}G}{d\nu ^{2}}}+\left(a-{\frac {c^{2}k^{2}}{2}}\cos 2\nu \right)G=0\\\end{aligned}}}</annotation>

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</math><img src="./a20fc963e996f8ed5cdf09e5373cb2a722c1263d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.539ex; margin-bottom: -0.299ex; width:35.226ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}&{\frac {d^{2}F}{d\mu ^{2}}}-\left(a-{\frac {c^{2}k^{2}}{2}}\cosh 2\mu \right)F=0\\&{\frac {d^{2}G}{d\nu ^{2}}}+\left(a-{\frac {c^{2}k^{2}}{2}}\cos 2\nu \right)G=0\\\end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is a separation constant.

As a specific physical example, the Helmholtz equation can be interpreted as describing <a href="Normal_mode" title="Normal mode">normal modes</a> of an elastic membrane under uniform <a href="Tension_(physics)" title="Tension (physics)">tension</a>. In this case, the following physical conditions are imposed:<a href="#cite_note-40">[40]</a>

<ul><li>Periodicity with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mu ,\nu )=\psi (\mu ,\nu +2\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mu ,\nu )=\psi (\mu ,\nu +2\pi )}</annotation>

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</math><img src="./aad9636d2a858f472ed97b1a8ed3dc13dc976294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.414ex; height:2.843ex;" alt="{\displaystyle \psi (\mu ,\nu )=\psi (\mu ,\nu +2\pi )}" loading="lazy"></li>

<li>Continuity of displacement across the interfocal line: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (0,\nu )=\psi (0,-\nu )}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (0,\nu )=\psi (0,-\nu )}</annotation>

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</math><img src="./34dd22c8a139f2027d91006952f3dfe1748a883f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.409ex; height:2.843ex;" alt="{\displaystyle \psi (0,\nu )=\psi (0,-\nu )}" loading="lazy"></li>

<li>Continuity of derivative across the interfocal line: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mu }(0,\nu )=-\psi _{\mu }(0,-\nu )}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\mu }(0,\nu )=-\psi _{\mu }(0,-\nu )}</annotation>

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</math><img src="./c9096b8c0924560a22ac9eff7ded02bab95e9603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.664ex; height:3.009ex;" alt="{\displaystyle \psi _{\mu }(0,\nu )=-\psi _{\mu }(0,-\nu )}" loading="lazy"></li></ul>

For given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, this restricts the solutions to those of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ce}}_{n}(\mu ,q){\text{ce}}_{n}(\nu ,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Ce}}_{n}(\mu ,q){\text{ce}}_{n}(\nu ,q)}</annotation>

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</math><img src="./8076fb4b0fc4d53bc7dbd585c12b436b9cb22dab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.672ex; height:2.843ex;" alt="{\displaystyle {\text{Ce}}_{n}(\mu ,q){\text{ce}}_{n}(\nu ,q)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Se}}_{n}(\mu ,q){\text{se}}_{n}(\nu ,q)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Se}}_{n}(\mu ,q){\text{se}}_{n}(\nu ,q)}</annotation>

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</math><img src="./5d0430e6e87766c080e864e6cfeabfb9e2077ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.17ex; height:2.843ex;" alt="{\displaystyle {\text{Se}}_{n}(\mu ,q){\text{se}}_{n}(\nu ,q)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=c^{2}k^{2}/4}">

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<annotation encoding="application/x-tex">{\displaystyle q=c^{2}k^{2}/4}</annotation>

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</math><img src="./37167293e104caf0c02975519ffb72a10829b8b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.819ex; height:3.176ex;" alt="{\displaystyle q=c^{2}k^{2}/4}" loading="lazy">. This is the same as restricting allowable values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, for given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. Restrictions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu _{0}>0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{0}>0}</annotation>

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</math><img src="./d355ad5234d6af4d65b89d15e687538994ec7176.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.217ex; height:2.676ex;" alt="{\displaystyle \mu =\mu _{0}>0}" loading="lazy">. For instance, clamping the membrane at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu _{0}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{0}}</annotation>

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</math><img src="./cd2d1d3ccf0ae20cb376782ebb04f8063c0a1c41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.956ex; height:2.176ex;" alt="{\displaystyle \mu =\mu _{0}}" loading="lazy"> imposes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mu _{0},\nu )=0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mu _{0},\nu )=0}</annotation>

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</math><img src="./735bf8a8b52393603d16e61276497b218de24ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.305ex; height:2.843ex;" alt="{\displaystyle \psi (\mu _{0},\nu )=0}" loading="lazy">, which in turn requires

<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{Ce}}_{n}(\mu _{0},q)=0\\{\text{Se}}_{n}(\mu _{0},q)=0\end{aligned}}}</annotation>

</semantics>

</math><img src="./404671b8e0e7c7202028d008d4bc3e522966fcce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.31ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}{\text{Ce}}_{n}(\mu _{0},q)=0\\{\text{Se}}_{n}(\mu _{0},q)=0\end{aligned}}}" loading="lazy"></dd></dl>

These conditions define the normal modes of the system.

<div class="mw-heading mw-heading3"><h3 id="Dynamical_problems">Dynamical problems</h3></div>

In dynamical problems with periodically varying forces, the <a href="Equation_of_motion" class="mw-redirect" title="Equation of motion">equation of motion</a> sometimes takes the form of Mathieu's equation. In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics.<a href="#cite_note-Meixner_applications-41">[41]</a> A classic example along these lines is the <a href="Inverted_pendulum" title="Inverted pendulum">inverted pendulum</a>.<a href="#cite_note-42">[42]</a> Other examples are

<ul><li>vibrations of a string with periodically varying tension<a href="#cite_note-Meixner_applications-41">[41]</a></li>

<li>stability of railroad rails as trains drive over them</li>

<li>seasonally forced <a href="Population_dynamics" title="Population dynamics">population dynamics</a></li>

<li>the phenomenon of <a href="Parametric_oscillator#Parametric_resonance" title="Parametric oscillator">parametric resonance</a> in forced <a href="Oscillator" class="mw-redirect" title="Oscillator">oscillators</a></li>

<li>motion of ions in a <a href="Quadrupole_ion_trap" title="Quadrupole ion trap">quadrupole ion trap</a><a href="#cite_note-43">[43]</a></li>

<li>the <a href="Stark_effect" title="Stark effect">Stark effect</a> for a rotating <a href="Electric_dipole" class="mw-redirect" title="Electric dipole">electric dipole</a></li>

<li>the <a href="Floquet_theory" title="Floquet theory">Floquet theory</a> of the stability of <a href="Limit_cycles" class="mw-redirect" title="Limit cycles">limit cycles</a></li>

<li>analytic <a href="Traveling_wave" class="mw-redirect" title="Traveling wave">traveling-wave</a> solutions of the <a href="Kardar%E2%80%93Parisi%E2%80%93Zhang_equation" title="Kardar–Parisi–Zhang equation">Kardar-Parisi-Zhang</a> interface growing equation with periodic noise term<a href="#cite_note-44">[44]</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3></div>

Mathieu functions play a role in certain quantum mechanical systems, particularly those with spatially periodic potentials such as the <a href="Quantum_pendulum" title="Quantum pendulum">quantum pendulum</a> and <a href="Crystalline_lattice" class="mw-redirect" title="Crystalline lattice">crystalline lattices</a>.

The modified Mathieu equation also arises when describing the quantum mechanics of singular potentials. For the particular singular potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(r)=g^{2}/r^{4}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle V(r)=g^{2}/r^{4}}</annotation>

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</math><img src="./a95505911b4efe85a7f34c62deebab3c27d2b259.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.181ex; height:3.176ex;" alt="{\displaystyle V(r)=g^{2}/r^{4}}" loading="lazy"> the radial <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>

<mfrac>

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<mfrac>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y}{dr^{2}}}+\left[k^{2}-{\frac {\ell (\ell +1)}{r^{2}}}-{\frac {g^{2}}{r^{4}}}\right]y=0}</annotation>

</semantics>

</math><img src="./38b42479beb04b88f1b9ba6de495a30ae845482a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.905ex; height:6.343ex;" alt="{\displaystyle {\frac {d^{2}y}{dr^{2}}}+\left[k^{2}-{\frac {\ell (\ell +1)}{r^{2}}}-{\frac {g^{2}}{r^{4}}}\right]y=0}" loading="lazy"></dd></dl>

can be converted into the equation

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\varphi }{dz^{2}}}+\left[2h^{2}\cosh 2z-\left(\ell +{\frac {1}{2}}\right)^{2}\right]\varphi =0.}</annotation>

</semantics>

</math><img src="./0bc1b644f273866a13075cb154c732e4c21e7653.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.103ex; height:7.509ex;" alt="{\displaystyle {\frac {d^{2}\varphi }{dz^{2}}}+\left[2h^{2}\cosh 2z-\left(\ell +{\frac {1}{2}}\right)^{2}\right]\varphi =0.}" loading="lazy"></dd></dl>

The transformation is achieved with the following substitutions

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle y=r^{1/2}\varphi ,r=\gamma e^{z},\gamma ={\frac {ig}{h}},h^{2}=ikg,h=e^{I\pi /4}(kg)^{1/2}.}</annotation>

</semantics>

</math><img src="./40c743f8d936f839625cd274f9c3e7c1a7c132b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.537ex; height:5.509ex;" alt="{\displaystyle y=r^{1/2}\varphi ,r=\gamma e^{z},\gamma ={\frac {ig}{h}},h^{2}=ikg,h=e^{I\pi /4}(kg)^{1/2}.}" loading="lazy"></dd></dl>

By solving the Schrödinger equation (for this particular potential) in terms of solutions of the modified Mathieu equation, scattering properties such as the <a href="S-matrix" title="S-matrix">S-matrix</a> and the <a href="Absorptance" title="Absorptance">absorptivity</a> can be obtained.<a href="#cite_note-45">[45]</a>

Originally the Schrödinger equation with cosine function was solved in 1928 by Strutt.<a href="#cite_note-46">[46]</a>

<ul><li><a href="Almost_Mathieu_operator" title="Almost Mathieu operator">Almost Mathieu operator</a></li>

<li><a href="Bessel_function" title="Bessel function">Bessel function</a></li>

<li><a href="Hill_differential_equation" title="Hill differential equation">Hill differential equation</a></li>

<li><a href="Inverted_pendulum" title="Inverted pendulum">Inverted pendulum</a></li>

<li><a href="Lam%C3%A9_function" title="Lamé function">Lamé function</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of mathematical functions</a></li>

<li><a href="Monochromatic_electromagnetic_plane_wave" title="Monochromatic electromagnetic plane wave">Monochromatic electromagnetic plane wave</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-columns references-column-width reflist-columns-2">

<li id="cite_note-1"><a href="#cite_ref-1">^</a> Mathieu (1868).

</li>

<li id="cite_note-MorseandFeshbach-2"><a href="#cite_ref-MorseandFeshbach_2-0">^</a> Morse and Feshbach (1953).

</li>

<li id="cite_note-BrimacombeCorlessZamir-3"><a href="#cite_ref-BrimacombeCorlessZamir_3-0">^</a> Brimacombe, Corless and Zamir (2021)

</li>

<li id="cite_note-Gutiérrez-Vega2015-4">^ <a href="#cite_ref-Gutiérrez-Vega2015_4-0">a</a> <a href="#cite_ref-Gutiérrez-Vega2015_4-1">b</a> Gutiérrez-Vega (2015).

</li>

<li id="cite_note-Arscott_chapIII-5">^ <a href="#cite_ref-Arscott_chapIII_5-0">a</a> <a href="#cite_ref-Arscott_chapIII_5-1">b</a> <a href="#cite_ref-Arscott_chapIII_5-2">c</a> <a href="#cite_ref-Arscott_chapIII_5-3">d</a> <a href="#cite_ref-Arscott_chapIII_5-4">e</a> Arscott (1964), chapter III

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> Arscott (1964) 43–44

</li>

<li id="cite_note-McLachlan_chapII-7">^ <a href="#cite_ref-McLachlan_chapII_7-0">a</a> <a href="#cite_ref-McLachlan_chapII_7-1">b</a> <a href="#cite_ref-McLachlan_chapII_7-2">c</a> McLachlan (1947), chapter II.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Arscott (1964); Iyanaga (1980); Gradshteyn (2007); This is also the normalization used by the <a href="Computer_algebra_system" title="Computer algebra system">computer algebra system</a> <a href="Maple_(software)" title="Maple (software)">Maple</a>.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> Arscott (1964), p. 29.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> It is not true, in general, that a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"> periodic function has the property <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x+\pi )=-y(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle y(x+\pi )=-y(x)}</annotation>

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</math><img src="./6e04fbdbcc7d0e9f061263c0d3c5948e472634d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.668ex; height:2.843ex;" alt="{\displaystyle y(x+\pi )=-y(x)}" loading="lazy">. However, this turns out to be true for functions which are solutions of Mathieu's equation.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> McLachlan (1951), pp. 141-157, 372

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> Arscott (1964), p. 34

</li>

<li id="cite_note-FOOTNOTEButikov2018-13"><a href="#cite_ref-FOOTNOTEButikov2018_13-0">^</a> <a href="#CITEREFButikov2018">Butikov 2018</a>.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> McLachlan (1947), p. 144

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> McLachlan (1947), p. 372

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> McLachlan (1947), p. 28

</li>

<li id="cite_note-Wimp1984-17">^ <a href="#cite_ref-Wimp1984_17-0">a</a> <a href="#cite_ref-Wimp1984_17-1">b</a> Wimp (1984), pp. 83-84

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> McLachlan (1947)

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> Chaos-Cador and Ley-Koo (2001)

</li>

<li id="cite_note-Temme2015-20">^ <a href="#cite_ref-Temme2015_20-0">a</a> <a href="#cite_ref-Temme2015_20-1">b</a> Temme (2015), p. 234

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> Müller-Kirsten (2012), pp. 420-428

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> Meixner and Schäfke (1954); McLachlan (1947)

</li>

<li id="cite_note-Malits2010-23">^ <a href="#cite_ref-Malits2010_23-0">a</a> <a href="#cite_ref-Malits2010_23-1">b</a> Malits (2010)

</li>

<li id="cite_note-Jin1996-24"><a href="#cite_ref-Jin1996_24-0">^</a> Jin and Zhang (1996)

</li>

<li id="cite_note-Van_BurenBoisvert2007-25"><a href="#cite_ref-Van_BurenBoisvert2007_25-0">^</a> Van Buren and Boisvert (2007)

</li>

<li id="cite_note-Bibby2013-26"><a href="#cite_ref-Bibby2013_26-0">^</a> Bibby and Peterson (2013)

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> Meixner and Schäfke (1954), p.134

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> McLachlan (1947), pp. 234–235

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> Gradshteyn (2007), p. 953

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> Arscott (1964), pp. 40-41

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> Gradshteyn (2007), pp. 763–765

</li>

<li id="cite_note-32"><a href="#cite_ref-32">^</a> Arscott (1964), p. 86

</li>

<li id="cite_note-McLachlan_chapXI-33"><a href="#cite_ref-McLachlan_chapXI_33-0">^</a> McLachlan (1947), chapter XI

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> McLachlan (1947), p. 237; Dingle and Müller (1962); Müller (1962); Dingle and Müller(1964)

</li>

<li id="cite_note-Dingle_and_Müller_1962-35">^ <a href="#cite_ref-Dingle_and_Müller_1962_35-0">a</a> <a href="#cite_ref-Dingle_and_Müller_1962_35-1">b</a> <a href="#cite_ref-Dingle_and_Müller_1962_35-2">c</a> Dingle and Müller (1962)

</li>

<li id="cite_note-Müller-Kirsten_2012-36">^ <a href="#cite_ref-Müller-Kirsten_2012_36-0">a</a> <a href="#cite_ref-Müller-Kirsten_2012_36-1">b</a> Müller-Kirsten (2012)

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> Bibby and Peterson (2013); Barakat (1963); Sebak and Shafai (1991); Kretzschmar (1970)

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> Solon et al (2015)

</li>

<li id="cite_note-39"><a href="#cite_ref-39">^</a> see Willatzen and Voon (2011), pp. 61–65

</li>

<li id="cite_note-40"><a href="#cite_ref-40">^</a> McLachlan (1947), pp. 294–297

</li>

<li id="cite_note-Meixner_applications-41">^ <a href="#cite_ref-Meixner_applications_41-0">a</a> <a href="#cite_ref-Meixner_applications_41-1">b</a> Meixner and Schäfke (1954), pp. 324–343

</li>

<li id="cite_note-42"><a href="#cite_ref-42">^</a> Ruby (1996)

</li>

<li id="cite_note-43"><a href="#cite_ref-43">^</a> March (1997)

</li>

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</style><cite id="CITEREFBarnaBognárMátyásGuedda2020" class="citation book cs1">Barna, Imre Ferenc; Bognár, G.; Mátyás, L.; Guedda, M.; Hriczó, K. (2020). "Analytic Traveling-Wave Solutions of the Kardar-Parisi-Zhang Interface Growing Equation with Different Kind of Noise Terms". In Pinelas, S.; Graef, John R.; Hilger, S.; Kloeden, P; Shinas, C. (eds.). Differential and Difference Equations with Applications, ICDDEA 2019, Lisbon, Portugal, July 1–5 Conference proceedings. Springer. pp. 239–254. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1908.09615">1908.09615</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783030563226</bdi>.</cite>

</li>

<li id="cite_note-45"><a href="#cite_ref-45">^</a> Müller-Kirsten (2006)

</li>

<li id="cite_note-46"><a href="#cite_ref-46">^</a> <cite id="CITEREFStrutt1928" class="citation journal cs1">Strutt, M.J.O. (1928). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/andp.19283911006">"Zur Wellenmechanik des Atomgitters"</a>. Annalen der Physik. 86 (10): 319–324. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1928AnP...391..319S">1928AnP...391..319S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19283911006">10.1002/andp.19283911006</a>.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em">

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<li><cite id="CITEREFDingleMüller1964" class="citation journal cs1">Dingle, Robert B.; Müller, Harald J.W. (1964). "The Form of the Coefficients of the Late Terms in Asymptotic Expansions of the Characteristic Numbers of Mathieu and Spheroidal-Wave Functions". <a href="Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a>. 216: 123–133. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>.</cite></li>

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<li><cite id="CITEREFGutiérrez-Vega2015" class="citation cs2">Gutiérrez-Vega, Julio C. (2015), "Mathieu Functions", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, pp. 159–160</cite></li>

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<li><cite id="CITEREFJinZhang1996" class="citation book cs1">Jin, J.M.; Zhang, Shan Jjie (1996). Computation of special functions. New York: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780471119630</bdi>.</cite></li>

<li><cite id="CITEREFKretzschmar1970" class="citation cs2">Kretzschmar, J.G. (1970), "Wave Propagation in Hollow Conducting Elliptical Waveguides", IEEE Transactions on Microwave Theory and Techniques, 18 (9): 547–554, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1970ITMTT..18..547K">1970ITMTT..18..547K</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTMTT.1970.1127288">10.1109/TMTT.1970.1127288</a></cite></li>

<li><cite id="CITEREFMalits2010" class="citation cs2">Malits, Pinchas (2010), "Relations between Mathieu functions of the first and second kind", Integral Transforms and Special Functions, 21 (6): 423–436, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10652460903360499">10.1080/10652460903360499</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122033386">122033386</a></cite></li>

<li><cite id="CITEREFMarch1997" class="citation journal cs1">March, Raymond E. (April 1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F%28SICI%291096-9888%28199704%2932%3A4%3C351%3A%3AAID-JMS512%3E3.0.CO%3B2-Y">"An Introduction to Quadrupole Ion Trap Mass Spectrometry"</a>. Journal of Mass Spectrometry. 32 (4): 351–369. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997JMSp...32..351M">1997JMSp...32..351M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F%28SICI%291096-9888%28199704%2932%3A4%3C351%3A%3AAID-JMS512%3E3.0.CO%3B2-Y">10.1002/(SICI)1096-9888(199704)32:4<351::AID-JMS512>3.0.CO;2-Y</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16506573">16506573</a>.</cite></li>

<li><cite id="CITEREFMathieu,_E.1868" class="citation cs2">Mathieu, E. (1868), <a rel="nofollow" class="external text" href="http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16412&Deb=145&Fin=211&E=PDF">"Mémoire sur Le Mouvement Vibratoire d'une Membrane de forme Elliptique"</a>, <a href="Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es" title="Journal de Mathématiques Pures et Appliquées">Journal de Mathématiques Pures et Appliquées</a>: 137–203</cite></li>

<li><cite id="CITEREFMcLachlan,_N._W.1951" class="citation book cs1">McLachlan, N. W. (1951). Theory and application of Mathieu functions. Oxford University Press.</cite> Note: Reprinted lithographically in Great Britain at the University Press, Oxford, 1951 from corrected sheets of the (1947) first edition.</li>

<li><cite id="CITEREFMeixnerSchäfke1954" class="citation book cs1 cs1-prop-foreign-lang-source">Meixner, Josef; Schäfke, Friedrich Wilhelm (1954). Mathieusche Funktionen und Sphäroidfunktionen (in German). Berlin: Springer-Verlag. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-00941-3">10.1007/978-3-662-00941-3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-01806-3</bdi>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)</li>

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<li><cite id="CITEREFMüller-Kirsten2012" class="citation book cs1">Müller-Kirsten, Harald J.W. (2012). Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral (2nd ed.). World Scientific. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-4397--73-5</bdi>.</cite></li>

<li><cite id="CITEREFDingleMüller1962" class="citation journal cs1">Dingle, R.B.; Müller, H.J.W. (1962). "Asymptotic Expansions of Mathieu Functions and their Characteristic Numbers". <a href="Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a>. 1962 (211): 11–32. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1962.211.11">10.1515/crll.1962.211.11</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117516747">117516747</a>.</cite></li>

<li><cite id="CITEREFMüller1962" class="citation journal cs1">Müller, H.J.W. (1962). "On Asymptotic Expansions of Mathieu Functions". <a href="Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a>. 1962 (211): 179–190. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1962.211.179">10.1515/crll.1962.211.179</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118909645">118909645</a>.</cite></li>

<li><cite id="CITEREFSebakShafai1991" class="citation cs2">Sebak, A.; Shafai, L. (1991), "Generalized solutions for electromagnetic scattering by elliptical structures", Computer Physics Communications, 68 (1–3): 315–330, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991CoPhC..68..315S">1991CoPhC..68..315S</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0010-4655%2891%2990206-Z">10.1016/0010-4655(91)90206-Z</a></cite></li>

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<li><cite id="CITEREFTemme2015" class="citation cs2">Temme, Nico M. (2015), "Special Functions", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, p. 234</cite></li>

<li><cite id="CITEREFVan_BurenBoisvert2007" class="citation journal cs1">Van Buren, Arnie L.; Boisvert, Jeffrey E. (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0033-569X-07-01039-5">"Accurate calculation of the modified Mathieu functions of integer order"</a>. Quarterly of Applied Mathematics. 65 (1): 1–23. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0033-569X-07-01039-5">10.1090/S0033-569X-07-01039-5</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-569X">0033-569X</a>.</cite></li>

<li><cite id="CITEREFLew_Yan_VoonWillatzen2011" class="citation book cs1">Lew Yan Voon LC, Willatzen M (2011). Separable Boundary-Value Problems in Physics. Wiley-VCH. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9783527634927">10.1002/9783527634927</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-527-41020-0</bdi>.</cite> (free online access to the appendix on Mathieu functions)</li>

<li><cite id="CITEREFWimp1984" class="citation book cs1">Wimp, Jet (1984). Computation with Recurrence Relations. Pitman Publishing. pp. 83–84. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-273-08508-5</bdi>.</cite></li>

<li><cite id="CITEREFWolf2010" class="citation cs2">Wolf, G. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/28">"Mathieu Functions and Hill's Equation"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</li>

<li><cite id="CITEREFBrimacombeCorlessZamir2021" class="citation journal cs1">Brimacombe, Chris; Corless, Robert M.; Zamir, Mair (2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F20M135786X">"Computation and Applications of Mathieu Functions: A Historical Perspective"</a>. SIAM Review. 63 (4): 653–720. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2008.01812">2008.01812</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F20M135786X">10.1137/20M135786X</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0036-1445">0036-1445</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220969117">220969117</a>.</cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/MathieuFunction.html">"Mathieu function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://functions.wolfram.com/MathieuandSpheroidalFunctions/">List of equations and identities for Mathieu Functions</a> functions.wolfram.com</li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Mathieu_functions">"Mathieu functions"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li>Timothy Jones, <a rel="nofollow" class="external text" href="http://www.physics.drexel.edu/~tim/open/mat/mat.html">Mathieu's Equations and the Ideal rf-Paul Trap</a> (2006)</li>

<li><a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/en/solutions/ode/ode0234.pdf">Mathieu equation</a>, <a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/en/">EqWorld</a></li>

<li><a rel="nofollow" class="external text" href="http://dlmf.nist.gov/28">NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation</a></li></ul>

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