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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Struve function</span></span>
</h1>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>Struve functions</b> <span class="texhtml"><b>H</b><sub><i>α</i></sub>(<i>x</i>)</span>, are solutions <span class="texhtml"><i>y</i>(<i>x</i>)</span> of the non-homogeneous <a href="Bessel's_differential_equation" class="mw-redirect" title="Bessel's differential equation">Bessel's differential equation</a>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>x</mi>
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<mo>−<!-- − --></mo>
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<mi>y</mi>
<mo>=</mo>
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<mn>4</mn>
<msup>
<mrow>
<mo>(</mo>
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<mi>x</mi>
<mn>2</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
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<mi>π<!-- π --></mi>
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}</annotation>
</semantics>
</math></span><img src="./246a7bab900d24f188a7edcec59042852728d747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.809ex; height:8.176ex;" alt="{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}" loading="lazy"></span></dd></dl>
<p>introduced by <a href="Hermann_Struve" class="mw-redirect" title="Hermann Struve">Hermann Struve</a>&nbsp;(<a href="#CITEREFStruve1882">1882</a>). The <a href="Complex_number" title="Complex number">complex number</a> α is the <b>order</b> of the Struve function, and is often an integer.
</p><p>And further defined its second-kind version <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} _{\alpha }(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">K</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {K} _{\alpha }(x)}</annotation>
</semantics>
</math></span><img src="./d06b60f8e3dc5d508921788a4c7311280fd7efca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.517ex; height:2.843ex;" alt="{\displaystyle \mathbf {K} _{\alpha }(x)}" loading="lazy"></span> as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">K</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
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<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
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</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
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</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)}</annotation>
</semantics>
</math></span><img src="./8cc05c041501cf89a2699f3654c00c86acb5db96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.744ex; height:2.843ex;" alt="{\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\alpha }(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{\alpha }(x)}</annotation>
</semantics>
</math></span><img src="./945231f316d299ffcff269a5d69e61b27a8088d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.774ex; height:2.843ex;" alt="{\displaystyle Y_{\alpha }(x)}" loading="lazy"></span> is the <a href="Bessel_function#Bessel_functions_of_the_second_kind_:_Y.CE.B1" title="Bessel function">Neumann function</a>.
</p><p>The <b>modified Struve functions</b> <span class="texhtml"><b>L</b><sub><i>α</i></sub>(<i>x</i>)</span> are equal to <span class="texhtml">−<i>ie</i><sup>−<i>iαπ</i> / 2</sup><b>H</b><sub><i>α</i></sub>(<i>ix</i>)</span> and are solutions <span class="texhtml"><i>y</i>(<i>x</i>)</span> of the non-homogeneous <a href="Bessel's_differential_equation" class="mw-redirect" title="Bessel's differential equation">Bessel's differential equation</a>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mi>y</mi>
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<mi>d</mi>
<msup>
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<mn>2</mn>
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<mo>+</mo>
<mi>x</mi>
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<mfrac>
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<mi>d</mi>
<mi>y</mi>
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<mi>d</mi>
<mi>x</mi>
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<mo>−<!-- − --></mo>
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<mo>(</mo>
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<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>+</mo>
<msup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>4</mn>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
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<mo>)</mo>
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</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}</annotation>
</semantics>
</math></span><img src="./c9e81a0e25186697d3ce13726811dada840628bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.809ex; height:8.176ex;" alt="{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}" loading="lazy"></span></dd></dl>
<p>And further defined its second-kind version <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{\alpha }(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">M</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{\alpha }(x)}</annotation>
</semantics>
</math></span><img src="./4867af824b58ee34cada5f406e362d8ed7d5132c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.961ex; height:2.843ex;" alt="{\displaystyle \mathbf {M} _{\alpha }(x)}" loading="lazy"></span> as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">M</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
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<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)}</annotation>
</semantics>
</math></span><img src="./9bad5cc5c5adb6fbcb262b85dfb8dc07a6e7daff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.377ex; height:2.843ex;" alt="{\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\alpha }(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{\alpha }(x)}</annotation>
</semantics>
</math></span><img src="./b89f5fe7690de97fb4cf2fb15028caacdcb61975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.446ex; height:2.843ex;" alt="{\displaystyle I_{\alpha }(x)}" loading="lazy"></span> is the <a href="Bessel_function#Bessel_functions_of_the_second_kind_:_Y.CE.B1" title="Bessel function">modified Bessel function</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>
<p>Since this is a <a href="Ordinary_differential_equation#homogeneous" title="Ordinary differential equation">non-homogeneous</a> equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the <a href="Bessel_function" title="Bessel function">Bessel functions</a>, and the particular solution may be chosen as the corresponding Struve function.
</p>
<div class="mw-heading mw-heading3"><h3 id="Power_series_expansion">Power series expansion</h3></div>
<p>Struve functions, denoted as <span class="texhtml"><b>H</b><sub><i>α</i></sub>(<i>z</i>)</span> have the <a href="Power_series" title="Power series">power series</a> form
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1},}">
<semantics>
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<mi mathvariant="bold">H</mi>
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<mi>α<!-- α --></mi>
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<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
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<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>m</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
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<mo>)</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>m</mi>
<mo>+</mo>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
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<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1},}</annotation>
</semantics>
</math></span><img src="./5756f6423f8f16e0958632c03199b4d6342c0a1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:53.511ex; height:8.343ex;" alt="{\displaystyle \mathbf {H} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1},}" loading="lazy"></span></dd></dl>
<p>where <span class="texhtml">Γ(<i>z</i>)</span> is the <a href="Gamma_function" title="Gamma function">gamma function</a>.
</p><p>The modified Struve functions, denoted <span class="texhtml"><b>L</b><sub><i>α</i></sub>(<i>z</i>)</span>, have the following power series form
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {1}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1}.}">
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<mi>α<!-- α --></mi>
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<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mo>=</mo>
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<mi mathvariant="normal">∞<!-- ∞ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>2</mn>
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<mo>)</mo>
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
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<mi>m</mi>
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<mrow>
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<mn>2</mn>
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<mn>1</mn>
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</msup>
<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {1}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1}.}</annotation>
</semantics>
</math></span><img src="./01de23167a8d22163131279561e64e502618ad8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:53.028ex; height:8.343ex;" alt="{\displaystyle \mathbf {L} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {1}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1}.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Integral_form">Integral form</h3></div>
<p>Another definition of the Struve function, for values of <span class="texhtml mvar" style="font-style:italic;">α</span> satisfying <span class="texhtml">Re(<i>α</i>) &gt; − <style data-mw-deduplicate="TemplateStyles:r1214402035">
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</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, is possible expressing in term of the Poisson's integral representation:
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}\sin xt~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sin(x\cos \tau )\sin ^{2\alpha }\tau ~d\tau ={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sin(x\sin \tau )\cos ^{2\alpha }\tau ~d\tau }">
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<mi>α<!-- α --></mi>
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</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
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<msup>
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<mi>α<!-- α --></mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>π<!-- π --></mi>
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
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<msubsup>
<mo>∫<!-- ∫ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
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</msup>
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<mrow class="MJX-TeXAtom-ORD">
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<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<mn>2</mn>
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<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>x</mi>
<mi>t</mi>
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<mi>t</mi>
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<msqrt>
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</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
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<mi>α<!-- α --></mi>
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<mrow class="MJX-TeXAtom-ORD">
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<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">)</mo>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>α<!-- α --></mi>
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<mi>d</mi>
<mi>τ<!-- τ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
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<msup>
<mrow>
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<mi>x</mi>
<mn>2</mn>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
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<mrow class="MJX-TeXAtom-ORD">
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<mn>2</mn>
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</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
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<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>α<!-- α --></mi>
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</msup>
<mo>⁡<!-- ⁡ --></mo>
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<mtext>&nbsp;</mtext>
<mi>d</mi>
<mi>τ<!-- τ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}\sin xt~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sin(x\cos \tau )\sin ^{2\alpha }\tau ~d\tau ={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sin(x\sin \tau )\cos ^{2\alpha }\tau ~d\tau }</annotation>
</semantics>
</math></span></span>
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {K} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\infty }(1+t^{2})^{\alpha -{\frac {1}{2}}}e^{-xt}~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\infty }e^{-x\sinh \tau }\cosh ^{2\alpha }\tau ~d\tau }">
<semantics>
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<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mo>=</mo>
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<msup>
<mrow>
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</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
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<mi>π<!-- π --></mi>
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
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<mrow class="MJX-TeXAtom-ORD">
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</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
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<mo stretchy="false">(</mo>
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<msup>
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</mrow>
</msup>
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<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
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</msup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>x</mi>
<mi>t</mi>
</mrow>
</msup>
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<mi>d</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
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</mrow>
</mfrac>
</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</msubsup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>x</mi>
<mi>sinh</mi>
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</msup>
<msup>
<mi>cosh</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>α<!-- α --></mi>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
<mtext>&nbsp;</mtext>
<mi>d</mi>
<mi>τ<!-- τ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {K} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\infty }(1+t^{2})^{\alpha -{\frac {1}{2}}}e^{-xt}~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\infty }e^{-x\sinh \tau }\cosh ^{2\alpha }\tau ~d\tau }</annotation>
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</math></span></span>
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}\sinh xt~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sinh(x\cos \tau )\sin ^{2\alpha }\tau ~d\tau ={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sinh(x\sin \tau )\cos ^{2\alpha }\tau ~d\tau }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
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<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
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<mrow>
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<msup>
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<mn>2</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
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<mrow class="MJX-TeXAtom-ORD">
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<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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</msubsup>
<mo stretchy="false">(</mo>
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<mo>−<!-- − --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
</msup>
<mi>sinh</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>x</mi>
<mi>t</mi>
<mtext>&nbsp;</mtext>
<mi>d</mi>
<mi>t</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mrow>
</msubsup>
<mi>sinh</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">)</mo>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>α<!-- α --></mi>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
<mtext>&nbsp;</mtext>
<mi>d</mi>
<mi>τ<!-- τ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mrow>
</msubsup>
<mi>sinh</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">)</mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>α<!-- α --></mi>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
<mtext>&nbsp;</mtext>
<mi>d</mi>
<mi>τ<!-- τ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}\sinh xt~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sinh(x\cos \tau )\sin ^{2\alpha }\tau ~d\tau ={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sinh(x\sin \tau )\cos ^{2\alpha }\tau ~d\tau }</annotation>
</semantics>
</math></span></span>
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{\alpha }(x)=-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}e^{-xt}~dt=-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}e^{-x\cos \tau }\sin ^{2\alpha }\tau ~d\tau =-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}e^{-x\sin \tau }\cos ^{2\alpha }\tau ~d\tau }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">M</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
</msup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>x</mi>
<mi>t</mi>
</mrow>
</msup>
<mtext>&nbsp;</mtext>
<mi>d</mi>
<mi>t</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mrow>
</msubsup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>x</mi>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
</mrow>
</msup>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>α<!-- α --></mi>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
<mtext>&nbsp;</mtext>
<mi>d</mi>
<mi>τ<!-- τ --></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mrow>
</msubsup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>x</mi>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
</mrow>
</msup>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>α<!-- α --></mi>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mi>τ<!-- τ --></mi>
<mtext>&nbsp;</mtext>
<mi>d</mi>
<mi>τ<!-- τ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{\alpha }(x)=-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}e^{-xt}~dt=-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}e^{-x\cos \tau }\sin ^{2\alpha }\tau ~d\tau =-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}e^{-x\sin \tau }\cos ^{2\alpha }\tau ~d\tau }</annotation>
</semantics>
</math></span></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Asymptotic_forms">Asymptotic forms</h2></div>
<p>For small <span class="texhtml mvar" style="font-style:italic;">x</span>, the power series expansion is given <a href="#Power_series_expansion">above</a>.
</p><p>For large <span class="texhtml mvar" style="font-style:italic;">x</span>, one obtains:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha -1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}+O\left(\left({\tfrac {x}{2}}\right)^{\alpha -3}\right),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mi>O</mi>
<mrow>
<mo>(</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha -1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}+O\left(\left({\tfrac {x}{2}}\right)^{\alpha -3}\right),}</annotation>
</semantics>
</math></span><img src="./6b15dd07d23fef724897c6225f7cdce2868b1f75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:48.388ex; height:8.176ex;" alt="{\displaystyle \mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha -1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}+O\left(\left({\tfrac {x}{2}}\right)^{\alpha -3}\right),}" loading="lazy"></span></dd></dl>
<p>where <span class="texhtml"><i>Y<sub>α</sub></i>(<i>x</i>)</span> is the <a href="Bessel_function#Bessel_functions_of_the_second_kind_:_Y.CE.B1" title="Bessel function">Neumann function</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<p>The Struve functions satisfy the following recurrence relations:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&amp;={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&amp;=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mi>α<!-- α --></mi>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msup>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>π<!-- π --></mi>
</msqrt>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">H</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&amp;={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&amp;=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./65cd5bc5fecaab1021fb0d2b8e9c3c8f07a767db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:55.369ex; height:18.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&amp;={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&amp;=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_functions">Relation to other functions</h2></div>
<p>Struve functions of integer order can be expressed in terms of <a href="Anger_function" title="Anger function">Weber functions</a> <span class="texhtml"><b>E</b><sub><i>n</i></sub></span> and vice versa: if <span class="texhtml mvar" style="font-style:italic;">n</span> is a non-negative integer then
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {E} _{n}(z)&amp;={\frac {1}{\pi }}\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\frac {\Gamma \left(k+{\frac {1}{2}}\right)\left({\frac {z}{2}}\right)^{n-2k-1}}{\Gamma \left(n-k+{\frac {1}{2}}\right)}}-\mathbf {H} _{n}(z),\\\mathbf {E} _{-n}(z)&amp;={\frac {(-1)^{n+1}}{\pi }}\sum _{k=0}^{\left\lceil {\frac {n-3}{2}}\right\rceil }{\frac {\Gamma (n-k-{\frac {1}{2}})\left({\frac {z}{2}}\right)^{-n+2k+1}}{\Gamma \left(k+{\frac {3}{2}}\right)}}-\mathbf {H} _{-n}(z).\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {E} _{n}(z)&amp;={\frac {1}{\pi }}\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\frac {\Gamma \left(k+{\frac {1}{2}}\right)\left({\frac {z}{2}}\right)^{n-2k-1}}{\Gamma \left(n-k+{\frac {1}{2}}\right)}}-\mathbf {H} _{n}(z),\\\mathbf {E} _{-n}(z)&amp;={\frac {(-1)^{n+1}}{\pi }}\sum _{k=0}^{\left\lceil {\frac {n-3}{2}}\right\rceil }{\frac {\Gamma (n-k-{\frac {1}{2}})\left({\frac {z}{2}}\right)^{-n+2k+1}}{\Gamma \left(k+{\frac {3}{2}}\right)}}-\mathbf {H} _{-n}(z).\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./748e6301650552bfc7cc3efb35a7b79df3c14d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:62.147ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {E} _{n}(z)&amp;={\frac {1}{\pi }}\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\frac {\Gamma \left(k+{\frac {1}{2}}\right)\left({\frac {z}{2}}\right)^{n-2k-1}}{\Gamma \left(n-k+{\frac {1}{2}}\right)}}-\mathbf {H} _{n}(z),\\\mathbf {E} _{-n}(z)&amp;={\frac {(-1)^{n+1}}{\pi }}\sum _{k=0}^{\left\lceil {\frac {n-3}{2}}\right\rceil }{\frac {\Gamma (n-k-{\frac {1}{2}})\left({\frac {z}{2}}\right)^{-n+2k+1}}{\Gamma \left(k+{\frac {3}{2}}\right)}}-\mathbf {H} _{-n}(z).\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Struve functions of order <span class="texhtml"><i>n</i> + <span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> where <span class="texhtml mvar" style="font-style:italic;">n</span> is an integer can be expressed in terms of elementary functions. In particular if <span class="texhtml mvar" style="font-style:italic;">n</span> is a non-negative integer then
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{-n-{\frac {1}{2}}}(z)=(-1)^{n}J_{n+{\frac {1}{2}}}(z),}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{-n-{\frac {1}{2}}}(z)=(-1)^{n}J_{n+{\frac {1}{2}}}(z),}</annotation>
</semantics>
</math></span><img src="./15b2b85a72dc419b12918b05760fbf969a2ab9a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.2ex; height:4.176ex;" alt="{\displaystyle \mathbf {H} _{-n-{\frac {1}{2}}}(z)=(-1)^{n}J_{n+{\frac {1}{2}}}(z),}" loading="lazy"></span></dd></dl>
<p>where the right hand side is a <a href="Spherical_Bessel_function" class="mw-redirect" title="Spherical Bessel function">spherical Bessel function</a>.
</p><p>Struve functions (of any order) can be expressed in terms of the <a href="Generalized_hypergeometric_function" title="Generalized hypergeometric function">generalized hypergeometric function</a> <span class="texhtml"><sub>1</sub><i>F</i><sub>2</sub></span>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} _{\alpha }(z)={\frac {z^{\alpha +1}}{2^{\alpha }{\sqrt {\pi }}\Gamma \left(\alpha +{\tfrac {3}{2}}\right)}}{}_{1}F_{2}\left(1;{\tfrac {3}{2}},\alpha +{\tfrac {3}{2}};-{\tfrac {z^{2}}{4}}\right).}">
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} _{\alpha }(z)={\frac {z^{\alpha +1}}{2^{\alpha }{\sqrt {\pi }}\Gamma \left(\alpha +{\tfrac {3}{2}}\right)}}{}_{1}F_{2}\left(1;{\tfrac {3}{2}},\alpha +{\tfrac {3}{2}};-{\tfrac {z^{2}}{4}}\right).}</annotation>
</semantics>
</math></span><img src="./f492320990e41a486023d1d3d250e18889d50ef7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:50.479ex; height:8.343ex;" alt="{\displaystyle \mathbf {H} _{\alpha }(z)={\frac {z^{\alpha +1}}{2^{\alpha }{\sqrt {\pi }}\Gamma \left(\alpha +{\tfrac {3}{2}}\right)}}{}_{1}F_{2}\left(1;{\tfrac {3}{2}},\alpha +{\tfrac {3}{2}};-{\tfrac {z^{2}}{4}}\right).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<p>The Struve and Weber functions were shown to have an application to beamforming in.,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and in describing the effect of confining interface on Brownian motion of colloidal particles at low Reynolds numbers.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">K. Buchanan, C. Flores, S. Wheeland, J. Jensen, D. Grayson and G. Huff, "Transmit beamforming for radar applications using circularly tapered random arrays," 2017 IEEE Radar Conference (RadarConf), 2017, pp. 0112-0117, doi: 10.1109/RADAR.2017.7944181</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">B. U. Felderhof, "Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion." The Journal of Physical Chemistry B 109.45, 2005, pp. 21406-21412</span>
</li>
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</style><cite id="CITEREFR._M._Aarts_and_Augustus_J._E._M._Janssen2003" class="citation journal cs1">R. M. Aarts and Augustus J. E. M. Janssen (2003). "Approximation of the Struve function <i>H</i><sub>1</sub> occurring in impedance calculations". <i>J. Acoust. Soc. Am</i>. <b>113</b> (5): <span class="nowrap">2635–</span>2637. <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/2003ASAJ..113.2635A">2003ASAJ..113.2635A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.1564019">10.1121/1.1564019</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a>&nbsp;<a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12765381">12765381</a>.</cite></li>
<li><cite id="CITEREFR._M._Aarts_and_Augustus_J._E._M._Janssen2016" class="citation journal cs1">R. M. Aarts and Augustus J. E. M. Janssen (2016). <a rel="nofollow" class="external text" href="https://research.tue.nl/nl/publications/efficient-approximation-of-the-struve-functions-hn-occurring-in-the-calculation-of-sound-radiation-quantaties(c68b8858-9c9d-4ff2-bf39-e888bb638527).html">"Efficient approximation of the Struve functions <i>H</i><sub><i>n</i></sub> occurring in the calculation of sound radiation quantities"</a>. <i>J. Acoust. Soc. Am</i>. <b>140</b> (6): <span class="nowrap">4154–</span>4160. <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/2016ASAJ..140.4154A">2016ASAJ..140.4154A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.4968792">10.1121/1.4968792</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a>&nbsp;<a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28040027">28040027</a>.</cite></li>
<li><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_496.htm">"Chapter 12"</a>. <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol.&nbsp;55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first&nbsp;ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p.&nbsp;496. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a>&nbsp;<a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a>&nbsp;<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a>&nbsp;<a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite></li>
<li><cite id="CITEREFIvanov2001" class="citation cs2">Ivanov, A. B. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Struve_function">"Struve function"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>
<li><cite id="CITEREFParis2010" class="citation cs2">Paris, R. B. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/11">"Struve function"</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.), <i><a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a>&nbsp;<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</li>
<li><cite id="CITEREFStruve1882" class="citation journal cs1">Struve, H. (1882). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1423790">"Beitrag zur Theorie der Diffraction an Fernröhren"</a>. <i>Annalen der Physik und Chemie</i>. <b>17</b> (13): <span class="nowrap">1008–</span>1016. <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/1882AnP...253.1008S">1882AnP...253.1008S</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.18822531319">10.1002/andp.18822531319</a>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://functions.wolfram.com/Bessel-TypeFunctions/StruveH/introductions/Struves/">Struve functions</a> at <a rel="nofollow" class="external text" href="http://functions.wolfram.com">the Wolfram functions site</a>.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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