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Bessel function

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Bessel functions are <a href="Mathematics" title="Mathematics">mathematical</a> <a href="Special_functions" title="Special functions">special functions</a> that commonly appear in problems involving <a href="Wave_motion" class="mw-redirect" title="Wave motion">wave motion</a>, <a href="Heat_conduction" class="mw-redirect" title="Heat conduction">heat conduction</a>, and other physical phenomena with <a href="Circular_symmetry" title="Circular symmetry">circular symmetry</a> or <a href="Cylindrical_symmetry" class="mw-redirect" title="Cylindrical symmetry">cylindrical symmetry</a>. They are named after the German astronomer and mathematician <a href="Friedrich_Bessel" class="mw-redirect" title="Friedrich Bessel">Friedrich Bessel</a>, who studied them systematically in 1824.<a href="#cite_note-:0-1">[1]</a>

Bessel functions are solutions to a particular type of <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a>: <math display="block" 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=0,}">

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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=0,}</annotation>

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

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

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is a number that determines the shape of the solution. This number is called the order of the Bessel function and can be any complex number. Although the same equation arises for both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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

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</math><img src="./6807df17ce845a127ca43612e2ffd5cf62b7adc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle -\alpha }" loading="lazy">, mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes.

The most important cases are when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is an integer, the resulting Bessel functions are often called cylinder functions or <a href="Cylindrical_harmonics" title="Cylindrical harmonics">cylindrical harmonics</a> because they naturally arise when solving problems (like Laplace's equation) in cylindrical coordinates. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is a half-integer, the solutions are called <a href="Spherical_Bessel_functions" class="mw-redirect" title="Spherical Bessel functions">spherical Bessel functions</a> and are used in spherical systems, such as in solving the <a href="Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a> in <a href="Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a>.

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

Bessel's equation arises when finding separable solutions to <a href="Laplace's_equation" title="Laplace's equation">Laplace's equation</a> and the <a href="Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a> in cylindrical or <a href="Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a>. Bessel functions are therefore especially important for many problems of <a href="Wave_propagation" class="mw-redirect" title="Wave propagation">wave propagation</a> and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n + 1/2). For example:

<ul><li><a href="Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic waves</a> in a cylindrical <a href="Waveguide" title="Waveguide">waveguide</a></li>

<li>Pressure amplitudes of <a href="Inviscid_flow" title="Inviscid flow">inviscid</a> rotational flows</li>

<li><a href="Conduction_(heat)" class="mw-redirect" title="Conduction (heat)">Heat conduction</a> in a cylindrical object</li>

<li>Modes of vibration of a thin circular or annular <a href="Acoustic_membrane" title="Acoustic membrane">acoustic membrane</a> (such as a <a href="Drumhead" title="Drumhead">drumhead</a> or other <a href="Membranophone" title="Membranophone">membranophone</a>) or thicker plates such as sheet metal (see <a href="Kirchhoff%E2%80%93Love_plate_theory" title="Kirchhoff–Love plate theory">Kirchhoff–Love plate theory</a>, <a href="Mindlin%E2%80%93Reissner_plate_theory" class="mw-redirect" title="Mindlin–Reissner plate theory">Mindlin–Reissner plate theory</a>)</li>

<li>Diffusion problems on a lattice</li>

<li>Solutions to the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> in spherical and cylindrical coordinates for a free particle</li>

<li>Position space representation of the Feynman <a href="Propagator" title="Propagator">propagator</a> in <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a></li>

<li>Solving for patterns of acoustical radiation</li>

<li>Frequency-dependent friction in circular pipelines</li>

<li>Dynamics of floating bodies</li>

<li><a href="Angular_resolution" title="Angular resolution">Angular resolution</a></li>

<li>Diffraction from helical objects, including <a href="DNA" title="DNA">DNA</a></li>

<li><a href="Probability_density_function" title="Probability density function">Probability density function</a> of product of two normally distributed random variables<a href="#cite_note-2">[2]</a></li>

<li>Analyzing of the surface waves generated by microtremors, in <a href="Geophysics" title="Geophysics">geophysics</a> and <a href="Seismology" title="Seismology">seismology</a>.</li></ul>

Bessel functions also appear in other problems, such as signal processing (e.g., see <a href="Frequency_modulation_synthesis" title="Frequency modulation synthesis">FM audio synthesis</a>, <a href="Kaiser_window" title="Kaiser window">Kaiser window</a>, or <a href="Bessel_filter" title="Bessel filter">Bessel filter</a>).

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

Because this is a <a href="Linear_differential_equation" title="Linear differential equation">linear differential equation</a>, solutions can be scaled to any amplitude. The amplitudes chosen for the functions originate from the early work in which the functions appeared as <a href="#Bessel's_integrals">solutions to definite integrals</a> rather than solutions to differential equations. Because the differential equation is second-order, there must be two <a href="Linear_independence" title="Linear independence">linearly independent</a> solutions: one of the first kind and one of the second kind. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.The subscript n is typically used in place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<th>First kind</th>

<th>Second kind

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<td>Bessel functions

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<td><a href="#Modified_Bessel_functions">Modified Bessel functions</a>

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<td>Iα

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<td>Kα

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<td><a href="#Hankel_functions">Hankel functions</a>

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<td>H(1) α = Jα + iYα

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<td>H(2) α = Jα − iYα

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<td><a href="#Spherical_Bessel_functions">Spherical Bessel functions</a>

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<td>Modified spherical Bessel functions

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<td><a href="#Spherical_Hankel_functions">Spherical Hankel functions</a>

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<td>h(1) n = jn + iyn

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<td>h(2) n = jn − iyn

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Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by Nn and nn, respectively, rather than Yn and yn.<a href="#cite_note-3">[3]</a><a href="#cite_note-MathWorld-4">[4]</a>

<div class="mw-heading mw-heading3"><h3 id="Bessel_functions_of_the_first_kind:_Jα">Bessel functions of the first kind: Jα </h3></div>

Bessel functions of the first kind, denoted as Jα(x), are solutions of Bessel's differential equation. For integer or positive α, Bessel functions of the first kind are finite at the origin (x = 0); while for negative non-integer α, Bessel functions of the first kind diverge as x approaches zero. It is possible to define the function by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\alpha }}">

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</math><img src="./c7ecb06c51aa5eb67cb4363751c8774977ba3cd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.614ex; height:2.343ex;" alt="{\displaystyle x^{\alpha }}" loading="lazy"> times a <a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a> (note that α need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying the <a href="Frobenius_method" title="Frobenius method">Frobenius method</a> to Bessel's equation:<a href="#cite_note-p360-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }(x)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha },}</annotation>

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where Γ(z) is the <a href="Gamma_function" title="Gamma function">gamma function</a>, a shifted generalization of the <a href="Factorial" title="Factorial">factorial</a> function to non-integer values. Some earlier authors define the Bessel function of the first kind differently, essentially without the division by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}">

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

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

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</math><img src="./c68203be699cce32f38b291115d5bc2c6220dc9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle x/2}" loading="lazy">;<a href="#cite_note-6">[6]</a> this definition is not used in this article. The Bessel function of the first kind is an <a href="Entire_function" title="Entire function">entire function</a> if α is an integer, otherwise it is a <a href="Multivalued_function" title="Multivalued function">multivalued function</a> with singularity at zero. The graphs of Bessel functions look roughly like oscillating <a href="Sine_function" class="mw-redirect" title="Sine function">sine</a> or <a href="Cosine" class="mw-redirect" title="Cosine">cosine</a> functions that decay proportionally to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-{1}/{2}}}">

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

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For non-integer α, the functions Jα(x) and J−α(x) are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order n, the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):<a href="#cite_note-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle J_{-n}(x)=(-1)^{n}J_{n}(x).}</annotation>

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This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

<div class="mw-heading mw-heading4"><h4 id="Bessel's_integrals">Bessel's integrals</h4></div>

Another definition of the Bessel function, for integer values of n, is possible using an integral representation:<a href="#cite_note-Temme-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle J_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(n\tau -x\sin \tau )\,d\tau ={\frac {1}{\pi }}\operatorname {Re} \left(\int _{0}^{\pi }e^{i(n\tau -x\sin \tau )}\,d\tau \right),}</annotation>

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which is also called Hansen-Bessel formula.<a href="#cite_note-9">[9]</a>

This was the approach that Bessel used,<a href="#cite_note-10">[10]</a> and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for Re(x) > 0:<a href="#cite_note-Temme-8">[8]</a><a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a>

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<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\alpha \tau -x\sin \tau )\,d\tau -{\frac {\sin(\alpha \pi )}{\pi }}\int _{0}^{\infty }e^{-x\sinh t-\alpha t}\,dt.}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Relation_to_hypergeometric_series">Relation to hypergeometric series</h4></div>

The Bessel functions can be expressed in terms of the <a href="Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">generalized hypergeometric series</a> as<a href="#cite_note-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}\;_{0}F_{1}\left(\alpha +1;-{\frac {x^{2}}{4}}\right).}</annotation>

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This expression is related to the development of Bessel functions in terms of the <a href="Bessel%E2%80%93Clifford_function" title="Bessel–Clifford function">Bessel–Clifford function</a>.

<div class="mw-heading mw-heading4"><h4 id="Relation_to_Laguerre_polynomials">Relation to Laguerre polynomials</h4></div>

In terms of the <a href="Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a> Lk and arbitrarily chosen parameter t, the Bessel function can be expressed as<a href="#cite_note-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {J_{\alpha }(x)}{\left({\frac {x}{2}}\right)^{\alpha }}}={\frac {e^{-t}}{\Gamma (\alpha +1)}}\sum _{k=0}^{\infty }{\frac {L_{k}^{(\alpha )}\left({\frac {x^{2}}{4t}}\right)}{\binom {k+\alpha }{k}}}{\frac {t^{k}}{k!}}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Bessel_functions_of_the_second_kind:_Yα">Bessel functions of the second kind: Yα </h3></div>

The Bessel functions of the second kind, denoted by Yα(x), occasionally denoted instead by Nα(x), are solutions of the Bessel differential equation that have a singularity at the origin (x = 0) and are <a href="Multivalued_function" title="Multivalued function">multivalued</a>. These are sometimes called Weber functions, as they were introduced by <a href="Heinrich_Martin_Weber" title="Heinrich Martin Weber">H. M. Weber</a> (<a href="#CITEREFWeber1873">1873</a>), and also Neumann functions after <a href="Carl_Neumann" title="Carl Neumann">Carl Neumann</a>.<a href="#cite_note-mhtlab.uwaterloo.ca-17">[17]</a>

For non-integer α, Yα(x) is related to Jα(x) by

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<annotation encoding="application/x-tex">{\displaystyle Y_{\alpha }(x)={\frac {J_{\alpha }(x)\cos(\alpha \pi )-J_{-\alpha }(x)}{\sin(\alpha \pi )}}.}</annotation>

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In the case of integer order n, the function is defined by taking the limit as a non-integer α tends to n:

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<annotation encoding="application/x-tex">{\displaystyle Y_{n}(x)=\lim _{\alpha \to n}Y_{\alpha }(x).}</annotation>

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If n is a nonnegative integer, we have the series<a href="#cite_note-18">[18]</a>

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<annotation encoding="application/x-tex">{\displaystyle Y_{n}(z)=-{\frac {\left({\frac {z}{2}}\right)^{-n}}{\pi }}\sum _{k=0}^{n-1}{\frac {(n-k-1)!}{k!}}\left({\frac {z^{2}}{4}}\right)^{k}+{\frac {2}{\pi }}J_{n}(z)\ln {\frac {z}{2}}-{\frac {\left({\frac {z}{2}}\right)^{n}}{\pi }}\sum _{k=0}^{\infty }(\psi (k+1)+\psi (n+k+1)){\frac {\left(-{\frac {z^{2}}{4}}\right)^{k}}{k!(n+k)!}}}</annotation>

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

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

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</math><img src="./0c02965f8dd8bfe2c0352b07c1193b8dc276c1d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.411ex; height:2.843ex;" alt="{\displaystyle \psi (z)}" loading="lazy"> is the <a href="Digamma_function" title="Digamma function">digamma function</a>, the <a href="Logarithmic_derivative" title="Logarithmic derivative">logarithmic derivative</a> of the <a href="Gamma_function" title="Gamma function">gamma function</a>.<a href="#cite_note-MathWorld-4">[4]</a>

There is also a corresponding integral formula (for Re(x) > 0):<a href="#cite_note-p._178-19">[19]</a>

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<annotation encoding="application/x-tex">{\displaystyle Y_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(x\sin \theta -n\theta )\,d\theta -{\frac {1}{\pi }}\int _{0}^{\infty }\left(e^{nt}+(-1)^{n}e^{-nt}\right)e^{-x\sinh t}\,dt.}</annotation>

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In the case where n = 0: (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> being <a href="Euler's_constant" title="Euler's constant">Euler's constant</a>)<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{0}\left(x\right)={\frac {4}{\pi ^{2}}}\int _{0}^{{\frac {1}{2}}\pi }\cos \left(x\cos \theta \right)\left(\gamma +\ln \left(2x\sin ^{2}\theta \right)\right)\,d\theta .}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{0}\left(x\right)={\frac {4}{\pi ^{2}}}\int _{0}^{{\frac {1}{2}}\pi }\cos \left(x\cos \theta \right)\left(\gamma +\ln \left(2x\sin ^{2}\theta \right)\right)\,d\theta .}</annotation>

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Yα(x) is necessary as the second linearly independent solution of the Bessel's equation when α is an integer. But Yα(x) has more meaning than that. It can be considered as a "natural" partner of Jα(x). See also the subsection on Hankel functions below.

When α is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

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<annotation encoding="application/x-tex">{\displaystyle Y_{-n}(x)=(-1)^{n}Y_{n}(x).}</annotation>

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Both Jα(x) and Yα(x) are <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a> of x on the <a href="Complex_plane" title="Complex plane">complex plane</a> cut along the negative real axis. When α is an integer, the Bessel functions J are <a href="Entire_function" title="Entire function">entire functions</a> of x. If x is held fixed at a non-zero value, then the Bessel functions are entire functions of α.

The Bessel functions of the second kind when α is an integer is an example of the second kind of solution in <a href="Fuchs's_theorem" title="Fuchs's theorem">Fuchs's theorem</a>.

<div class="mw-heading mw-heading3"><h3 id="Hankel_functions:_H(1)α,_H(2)α">Hankel functions: H(1) α, H(2) α </h3></div>

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, H(1) α(x) and H(2) α(x), defined as<a href="#cite_note-20">[20]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&=J_{\alpha }(x)+iY_{\alpha }(x),\\[5pt]H_{\alpha }^{(2)}(x)&=J_{\alpha }(x)-iY_{\alpha }(x),\end{aligned}}}</annotation>

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where i is the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. They are named after <a href="Hermann_Hankel" title="Hermann Hankel">Hermann Hankel</a>.

These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form ei f(x). For real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

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</math><img src="./80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\alpha }(x)}">

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</math><img src="./d615d7480e166b510bda8f25fef612ec08f87a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.713ex; height:2.843ex;" alt="{\displaystyle J_{\alpha }(x)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\alpha }(x)}">

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

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</math><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"> are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of <a href="Euler's_formula" title="Euler's formula">Euler's formula</a>, substituting H(1) α(x), H(2) α(x) for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\pm ix}}">

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</math><img src="./86fc52df7377e91fde1184f462ffbf0d1d257507.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.102ex; height:2.676ex;" alt="{\displaystyle e^{\pm ix}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\alpha }(x)}">

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

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

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</math><img src="./eb9af7ed6f44822021b74bb82b431022c7fd66b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.25ex; height:2.843ex;" alt="{\displaystyle \cos(x)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)}">

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

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</math><img src="./3a990a5545cac26c1c6821dca95d898bc80fe3a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.995ex; height:2.843ex;" alt="{\displaystyle \sin(x)}" loading="lazy">, as explicitly shown in the <a href="#Asymptotic_forms">asymptotic expansion</a>.

The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the <a href="Sign_convention" title="Sign convention">sign convention</a> for the <a href="Frequency" title="Frequency">frequency</a>).

Using the previous relationships, they can be expressed as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&={\frac {J_{-\alpha }(x)-e^{-\alpha \pi i}J_{\alpha }(x)}{i\sin \alpha \pi }},\\[5pt]H_{\alpha }^{(2)}(x)&={\frac {J_{-\alpha }(x)-e^{\alpha \pi i}J_{\alpha }(x)}{-i\sin \alpha \pi }}.\end{aligned}}}</annotation>

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If α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not:<a href="#cite_note-21">[21]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{-\alpha }^{(1)}(x)&=e^{\alpha \pi i}H_{\alpha }^{(1)}(x),\\[6mu]H_{-\alpha }^{(2)}(x)&=e^{-\alpha \pi i}H_{\alpha }^{(2)}(x).\end{aligned}}}</annotation>

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In particular, if α = m + <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠1/2⁠ with m a nonnegative integer, the above relations imply directly that

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{-(m+{\frac {1}{2}})}(x)&=(-1)^{m+1}Y_{m+{\frac {1}{2}}}(x),\\[5pt]Y_{-(m+{\frac {1}{2}})}(x)&=(-1)^{m}J_{m+{\frac {1}{2}}}(x).\end{aligned}}}</annotation>

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These are useful in developing the spherical Bessel functions (see below).

The Hankel functions admit the following integral representations for Re(x) > 0:<a href="#cite_note-22">[22]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&={\frac {1}{\pi i}}\int _{-\infty }^{+\infty +\pi i}e^{x\sinh t-\alpha t}\,dt,\\[5pt]H_{\alpha }^{(2)}(x)&=-{\frac {1}{\pi i}}\int _{-\infty }^{+\infty -\pi i}e^{x\sinh t-\alpha t}\,dt,\end{aligned}}}</annotation>

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where the integration limits indicate integration along a <a href="Methods_of_contour_integration" class="mw-redirect" title="Methods of contour integration">contour</a> that can be chosen as follows: from −∞ to 0 along the negative real axis, from 0 to ±πi along the imaginary axis, and from ±πi to +∞ ± πi along a contour parallel to the real axis.<a href="#cite_note-p._178-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="Modified_Bessel_functions:_Iα,_Kα">Modified Bessel functions: Iα, Kα </h3></div>

The Bessel functions are valid even for <a href="Complex_number" title="Complex number">complex</a> arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as<a href="#cite_note-23">[23]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{\alpha }(x)&=i^{-\alpha }J_{\alpha }(ix)=\sum _{m=0}^{\infty }{\frac {1}{m!\,\Gamma (m+\alpha +1)}}\left({\frac {x}{2}}\right)^{2m+\alpha },\\[5pt]K_{\alpha }(x)&={\frac {\pi }{2}}{\frac {I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi }},\end{aligned}}}</annotation>

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when α is not an integer. When α is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments x. The series expansion for Iα(x) is thus similar to that for Jα(x), but without the alternating (−1)m factor.

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

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<annotation encoding="application/x-tex">{\displaystyle K_{\alpha }(x)={\begin{cases}{\frac {\pi }{2}}i^{\alpha +1}H_{\alpha }^{(1)}(ix)&-\pi <\arg x\leq {\frac {\pi }{2}}\\{\frac {\pi }{2}}(-i)^{\alpha +1}H_{\alpha }^{(2)}(-ix)&-{\frac {\pi }{2}}<\arg x\leq \pi \end{cases}}}</annotation>

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Using these two formulae the result to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\alpha }^{2}(z)}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{\alpha }^{2}(z)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }^{2}(x)+Y_{\alpha }^{2}(x)={\frac {8}{\pi ^{2}}}\int _{0}^{\infty }\cosh(2\alpha t)K_{0}(2x\sinh t)\,dt,}</annotation>

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given that the condition Re(x) > 0 is met. It can also be shown that

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<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }^{2}(x)+Y_{\alpha }^{2}(x)={\frac {8\cos(\alpha \pi )}{\pi ^{2}}}\int _{0}^{\infty }K_{2\alpha }(2x\sinh t)\,dt,}</annotation>

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only when |Re(α)| < ⁠1/2⁠ and Re(x) ≥ 0 but not when x = 0.<a href="#cite_note-24">[24]</a>

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if −π < arg z ≤ ⁠π/2⁠):<a href="#cite_note-25">[25]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{\alpha }(iz)&=e^{\frac {\alpha \pi i}{2}}I_{\alpha }(z),\\[1ex]Y_{\alpha }(iz)&=e^{\frac {(\alpha +1)\pi i}{2}}I_{\alpha }(z)-{\tfrac {2}{\pi }}e^{-{\frac {\alpha \pi i}{2}}}K_{\alpha }(z).\end{aligned}}}</annotation>

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Iα(x) and Kα(x) are the two linearly independent solutions to the modified Bessel's equation:<a href="#cite_note-26">[26]</a>

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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=0.}</annotation>

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Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Iα and Kα are <a href="Exponential_growth" title="Exponential growth">exponentially growing</a> and <a href="Exponential_decay" title="Exponential decay">decaying</a> functions respectively. Like the ordinary Bessel function Jα, the function Iα goes to zero at x = 0 for α > 0 and is finite at x = 0 for α = 0. Analogously, Kα diverges at x = 0 with the singularity being of logarithmic type for K0, and ⁠1/2⁠Γ(|α|)(2/x)|α| otherwise.<a href="#cite_note-27">[27]</a>

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Two integral formulas for the modified Bessel functions are (for Re(x) > 0):<a href="#cite_note-28">[28]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{\alpha }(x)&={\frac {1}{\pi }}\int _{0}^{\pi }e^{x\cos \theta }\cos \alpha \theta \,d\theta -{\frac {\sin \alpha \pi }{\pi }}\int _{0}^{\infty }e^{-x\cosh t-\alpha t}\,dt,\\[5pt]K_{\alpha }(x)&=\int _{0}^{\infty }e^{-x\cosh t}\cosh \alpha t\,dt.\end{aligned}}}</annotation>

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Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for Re(ω) > 0):

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<annotation encoding="application/x-tex">{\displaystyle 2\,K_{0}(\omega )=\int _{-\infty }^{\infty }{\frac {e^{i\omega t}}{\sqrt {t^{2}+1}}}\,dt.}</annotation>

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It can be proven by showing equality to the above integral definition for K0. This is done by integrating a closed curve in the first quadrant of the complex plane.

Modified Bessel functions of the second kind may be represented with Bassett's integral <a href="#cite_note-29">[29]</a>

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<annotation encoding="application/x-tex">{\displaystyle K_{n}(xz)={\frac {\Gamma \left(n+{\frac {1}{2}}\right)(2z)^{n}}{{\sqrt {\pi }}x^{n}}}\int _{0}^{\infty }{\frac {\cos(xt)\,dt}{(t^{2}+z^{2})^{n+{\frac {1}{2}}}}}.}</annotation>

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Modified Bessel functions K1/3 and K2/3 can be represented in terms of rapidly convergent integrals<a href="#cite_note-30">[30]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K_{\frac {1}{3}}(\xi )&={\sqrt {3}}\int _{0}^{\infty }\exp \left(-\xi \left(1+{\frac {4x^{2}}{3}}\right){\sqrt {1+{\frac {x^{2}}{3}}}}\right)\,dx,\\[5pt]K_{\frac {2}{3}}(\xi )&={\frac {1}{\sqrt {3}}}\int _{0}^{\infty }{\frac {3+2x^{2}}{\sqrt {1+{\frac {x^{2}}{3}}}}}\exp \left(-\xi \left(1+{\frac {4x^{2}}{3}}\right){\sqrt {1+{\frac {x^{2}}{3}}}}\right)\,dx.\end{aligned}}}</annotation>

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The modified Bessel function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\frac {1}{2}}(\xi )=(2\xi /\pi )^{-1/2}\exp(-\xi )}">

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<annotation encoding="application/x-tex">{\displaystyle K_{\frac {1}{2}}(\xi )=(2\xi /\pi )^{-1/2}\exp(-\xi )}</annotation>

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</math><img src="./cc5deb67691e30d7f4ec81148813e3a60cbb3a7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.707ex; height:4.676ex;" alt="{\displaystyle K_{\frac {1}{2}}(\xi )=(2\xi /\pi )^{-1/2}\exp(-\xi )}" loading="lazy"> is useful to represent the <a href="Laplace_distribution" title="Laplace distribution">Laplace distribution</a> as an Exponential-scale mixture of normal distributions.

The modified Bessel function of the second kind has also been called by the following names (now rare):

<ul><li>Basset function after <a href="Alfred_Barnard_Basset" title="Alfred Barnard Basset">Alfred Barnard Basset</a></li>

<li>Modified Bessel function of the third kind</li>

<li>Modified Hankel function<a href="#cite_note-31">[31]</a></li>

<li>Macdonald function after <a href="Hector_Munro_Macdonald" title="Hector Munro Macdonald">Hector Munro Macdonald</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Spherical_Bessel_functions:_jn,_yn">Spherical Bessel functions: jn, yn </h3></div>

When solving the <a href="Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a> in spherical coordinates by <a href="Separation_of_variables#pde" title="Separation of variables">separation of variables</a>, the radial equation has the form

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<annotation encoding="application/x-tex">{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}+\left(x^{2}-n(n+1)\right)y=0.}</annotation>

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The two linearly independent solutions to this equation are called the spherical Bessel functions jn and yn, and are related to the ordinary Bessel functions Jn and Yn by<a href="#cite_note-32">[32]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}j_{n}(x)&={\sqrt {\frac {\pi }{2x}}}J_{n+{\frac {1}{2}}}(x),\\y_{n}(x)&={\sqrt {\frac {\pi }{2x}}}Y_{n+{\frac {1}{2}}}(x)=(-1)^{n+1}{\sqrt {\frac {\pi }{2x}}}J_{-n-{\frac {1}{2}}}(x).\end{aligned}}}</annotation>

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yn is also denoted nn or <a href="Eta_(letter)" class="mw-redirect" title="Eta (letter)">η</a>n; some authors call these functions the spherical Neumann functions.

From the relations to the ordinary Bessel functions it is directly seen that:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}j_{n}(x)&=(-1)^{n}y_{-n-1}(x)\\y_{n}(x)&=(-1)^{n+1}j_{-n-1}(x)\end{aligned}}}</annotation>

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The spherical Bessel functions can also be written as (<style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>Rayleigh's formulas)<a href="#cite_note-33">[33]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}j_{n}(x)&=(-x)^{n}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{n}{\frac {\sin x}{x}},\\y_{n}(x)&=-(-x)^{n}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{n}{\frac {\cos x}{x}}.\end{aligned}}}</annotation>

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The zeroth spherical Bessel function j0(x) is also known as the (unnormalized) <a href="Sinc_function" title="Sinc function">sinc function</a>. The first few spherical Bessel functions are:<a href="#cite_note-34">[34]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}j_{0}(x)&={\frac {\sin x}{x}}.\\j_{1}(x)&={\frac {\sin x}{x^{2}}}-{\frac {\cos x}{x}},\\j_{2}(x)&=\left({\frac {3}{x^{2}}}-1\right){\frac {\sin x}{x}}-{\frac {3\cos x}{x^{2}}},\\j_{3}(x)&=\left({\frac {15}{x^{3}}}-{\frac {6}{x}}\right){\frac {\sin x}{x}}-\left({\frac {15}{x^{2}}}-1\right){\frac {\cos x}{x}}\end{aligned}}}</annotation>

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and<a href="#cite_note-35">[35]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{0}(x)&=-j_{-1}(x)=-{\frac {\cos x}{x}},\\y_{1}(x)&=j_{-2}(x)=-{\frac {\cos x}{x^{2}}}-{\frac {\sin x}{x}},\\y_{2}(x)&=-j_{-3}(x)=\left(-{\frac {3}{x^{2}}}+1\right){\frac {\cos x}{x}}-{\frac {3\sin x}{x^{2}}},\\y_{3}(x)&=j_{-4}(x)=\left(-{\frac {15}{x^{3}}}+{\frac {6}{x}}\right){\frac {\cos x}{x}}-\left({\frac {15}{x^{2}}}-1\right){\frac {\sin x}{x}}.\end{aligned}}}</annotation>

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The first few non-zero roots of the first few spherical Bessel functions are:

<caption>Non-zero Roots of the Spherical Bessel Function (first kind)

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<th>Order</th>

<th>Root 5

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

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</math><img src="./a7829ecaa6c8e08ce2a635f4d0232b6ac68a81f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.039ex; height:2.509ex;" alt="{\displaystyle j_{0}}" loading="lazy"></td>

<td>15.707963

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

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</math><img src="./0a671019a672cd149a8d1584861e2f8fe9728a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.039ex; height:2.509ex;" alt="{\displaystyle j_{1}}" loading="lazy"></td>

<td>17.220755

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

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</math><img src="./9f29268662641c1e61d6faba764a493c74ba962b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.039ex; height:2.509ex;" alt="{\displaystyle j_{2}}" loading="lazy"></td>

<td>18.689036

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<annotation encoding="application/x-tex">{\displaystyle j_{3}}</annotation>

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</math><img src="./5b579a8d652b45503a5db2ac2d68118eebd6d5a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.039ex; height:2.509ex;" alt="{\displaystyle j_{3}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle j_{4}}</annotation>

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</math><img src="./e881f6b5cdd6fffc675ca244ccdcba2d0240975b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.039ex; height:2.509ex;" alt="{\displaystyle j_{4}}" loading="lazy"></td>

<td>21.525418

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<caption>Non-zero Roots of the Spherical Bessel Function (second kind)

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

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</math><img src="./6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"></td>

<td>14.137167

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eef4db76d658a98219aca14df06d9869d2b43c42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{1}}" loading="lazy"></td>

<td>15.644128

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7377c7399e662562cd420fa5c7ce49cfba574998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{2}}" loading="lazy"></td>

<td>17.103359

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{3}}</annotation>

</semantics>

</math><img src="./4e73a90104a9b6484a6bc2df35edf469d6307b2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{3}}" loading="lazy"></td>

<td>18.525210

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{4}}</annotation>

</semantics>

</math><img src="./6859e9003d906e626edc77215dd6a9a896fa3b1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{4}}" loading="lazy"></td>

<td>19.916796

</td></tr></tbody></table>

<div class="mw-heading mw-heading4"><h4 id="Generating_function">Generating function</h4></div>

The spherical Bessel functions have the generating functions<a href="#cite_note-36">[36]</a>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{z}}\cos \left({\sqrt {z^{2}-2zt}}\right)&=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}j_{n-1}(z),\\{\frac {1}{z}}\sin \left({\sqrt {z^{2}-2zt}}\right)&=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}y_{n-1}(z).\end{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Finite_series_expansions">Finite series expansions</h4></div>

In contrast to the whole integer Bessel functions Jn(x), Yn(x), the spherical Bessel functions jn(x), yn(x) have a finite series expression:<a href="#cite_note-37">[37]</a>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}j_{n}(x)&={\sqrt {\frac {\pi }{2x}}}J_{n+{\frac {1}{2}}}(x)=\\&={\frac {1}{2x}}\left[e^{ix}\sum _{r=0}^{n}{\frac {i^{r-n-1}(n+r)!}{r!(n-r)!(2x)^{r}}}+e^{-ix}\sum _{r=0}^{n}{\frac {(-i)^{r-n-1}(n+r)!}{r!(n-r)!(2x)^{r}}}\right]\\&={\frac {1}{x}}\left[\sin \left(x-{\frac {n\pi }{2}}\right)\sum _{r=0}^{\left[{\frac {n}{2}}\right]}{\frac {(-1)^{r}(n+2r)!}{(2r)!(n-2r)!(2x)^{2r}}}+\cos \left(x-{\frac {n\pi }{2}}\right)\sum _{r=0}^{\left[{\frac {n-1}{2}}\right]}{\frac {(-1)^{r}(n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}}}\right]\\y_{n}(x)&=(-1)^{n+1}j_{-n-1}(x)=(-1)^{n+1}{\frac {\pi }{2x}}J_{-\left(n+{\frac {1}{2}}\right)}(x)=\\&={\frac {(-1)^{n+1}}{2x}}\left[e^{ix}\sum _{r=0}^{n}{\frac {i^{r+n}(n+r)!}{r!(n-r)!(2x)^{r}}}+e^{-ix}\sum _{r=0}^{n}{\frac {(-i)^{r+n}(n+r)!}{r!(n-r)!(2x)^{r}}}\right]=\\&={\frac {(-1)^{n+1}}{x}}\left[\cos \left(x+{\frac {n\pi }{2}}\right)\sum _{r=0}^{\left[{\frac {n}{2}}\right]}{\frac {(-1)^{r}(n+2r)!}{(2r)!(n-2r)!(2x)^{2r}}}-\sin \left(x+{\frac {n\pi }{2}}\right)\sum _{r=0}^{\left[{\frac {n-1}{2}}\right]}{\frac {(-1)^{r}(n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}}}\right]\end{alignedat}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Differential_relations">Differential relations</h4></div>

In the following, fn is any of jn, yn, h(1) n, h(2) n for n = 0, ±1, ±2, ...<a href="#cite_note-38">[38]</a>

<mtr>

<mtd>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left({\frac {1}{z}}{\frac {d}{dz}}\right)^{m}\left(z^{n+1}f_{n}(z)\right)&=z^{n-m+1}f_{n-m}(z),\\\left({\frac {1}{z}}{\frac {d}{dz}}\right)^{m}\left(z^{-n}f_{n}(z)\right)&=(-1)^{m}z^{-n-m}f_{n+m}(z).\end{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Spherical_Hankel_functions:_h(1)n,_h(2)n">Spherical Hankel functions: h(1) n, h(2) n </h3></div>

There are also spherical analogues of the <a href="#Hankel_functions">Hankel functions</a>:

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{n}^{(1)}(x)&=j_{n}(x)+iy_{n}(x),\\h_{n}^{(2)}(x)&=j_{n}(x)-iy_{n}(x).\end{aligned}}}</annotation>

</semantics>

</math>

There are simple closed-form expressions for the Bessel functions of <a href="Half-integer" title="Half-integer">half-integer</a> order in terms of the standard <a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>, and therefore for the spherical Bessel functions. In particular, for non-negative integers n:

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{n}^{(1)}(x)=(-i)^{n+1}{\frac {e^{ix}}{x}}\sum _{m=0}^{n}{\frac {i^{m}}{m!\,(2x)^{m}}}{\frac {(n+m)!}{(n-m)!}},}</annotation>

</semantics>

</math>

and h(2) n is the complex-conjugate of this (for real x). It follows, for example, that j0(x) = ⁠sin x/x⁠ and y0(x) = −⁠cos x/x⁠, and so on.

The spherical Hankel functions appear in problems involving <a href="Spherical_wave" class="mw-redirect" title="Spherical wave">spherical wave</a> propagation, for example in the <a href="Electromagnetic_wave_equation#Multipole_expansion" title="Electromagnetic wave equation">multipole expansion of the electromagnetic field</a>.

<div class="mw-heading mw-heading3"><h3 id="Riccati–Bessel_functions:_Sn,_Cn,_ξn,_ζn">Riccati–Bessel functions: Sn, Cn, ξn, ζn </h3></div>

<a href="Jacopo_Riccati" title="Jacopo Riccati">Riccati</a>–Bessel functions only slightly differ from spherical Bessel functions:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{n}(x)&=xj_{n}(x)={\sqrt {\frac {\pi x}{2}}}J_{n+{\frac {1}{2}}}(x)\\C_{n}(x)&=-xy_{n}(x)=-{\sqrt {\frac {\pi x}{2}}}Y_{n+{\frac {1}{2}}}(x)\\\xi _{n}(x)&=xh_{n}^{(1)}(x)={\sqrt {\frac {\pi x}{2}}}H_{n+{\frac {1}{2}}}^{(1)}(x)=S_{n}(x)-iC_{n}(x)\\\zeta _{n}(x)&=xh_{n}^{(2)}(x)={\sqrt {\frac {\pi x}{2}}}H_{n+{\frac {1}{2}}}^{(2)}(x)=S_{n}(x)+iC_{n}(x)\end{aligned}}}</annotation>

</semantics>

</math>

They satisfy the differential equation

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+\left(x^{2}-n(n+1)\right)y=0.}</annotation>

</semantics>

</math>

For example, this kind of differential equation appears in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> while solving the radial component of the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> with hypothetical cylindrical infinite potential barrier.<a href="#cite_note-39">[39]</a> This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known as <a href="Mie_scattering" title="Mie scattering">Mie scattering</a> after the first published solution by Mie (1908). See e.g., Du (2004)<a href="#cite_note-40">[40]</a> for recent developments and references.

Following <a href="Peter_Debye" title="Peter Debye">Debye</a> (1909), the notation ψn, χn is sometimes used instead of Sn, Cn.

<div class="mw-heading mw-heading2"><h2 id="Asymptotic_forms">Asymptotic forms</h2></div>

The Bessel functions have the following <a href="Asymptotic_analysis" title="Asymptotic analysis">asymptotic</a> forms. For small arguments <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<z\ll {\sqrt {\alpha +1}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<z\ll {\sqrt {\alpha +1}}}</annotation>

</semantics>

</math><img src="./744ac401715e3d75c8ea69db0678eda9b3962543.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.389ex; height:3.009ex;" alt="{\displaystyle 0<z\ll {\sqrt {\alpha +1}}}" loading="lazy">, one obtains, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }(z)\sim {\frac {1}{\Gamma (\alpha +1)}}\left({\frac {z}{2}}\right)^{\alpha }.}</annotation>

</semantics>

</math>

When α is a negative integer, we have

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }(z)\sim {\frac {(-1)^{\alpha }}{(-\alpha )!}}\left({\frac {2}{z}}\right)^{\alpha }.}</annotation>

</semantics>

</math>

For the Bessel function of the second kind we have three cases:

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

<mtext> is a positive integer (one term dominates unless </mtext>

</mrow>

<mtext> is imaginary)</mtext>

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</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

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<msup>

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</msup>

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<msup>

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</mfrac>

</mstyle>

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</mrow>

</msup>

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<mtd>

</mrow>

<mtext> is a negative integer,</mtext>

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</mtd>

</mtr>

</mtable>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{\alpha }(z)\sim {\begin{cases}{\dfrac {2}{\pi }}\left(\ln \left({\dfrac {z}{2}}\right)+\gamma \right)&{\text{if }}\alpha =0\\[1ex]-{\dfrac {\Gamma (\alpha )}{\pi }}\left({\dfrac {2}{z}}\right)^{\alpha }+{\dfrac {1}{\Gamma (\alpha +1)}}\left({\dfrac {z}{2}}\right)^{\alpha }\cot(\alpha \pi )&{\text{if }}\alpha {\text{ is a positive integer (one term dominates unless }}\alpha {\text{ is imaginary)}},\\[1ex]-{\dfrac {(-1)^{\alpha }\Gamma (-\alpha )}{\pi }}\left({\dfrac {z}{2}}\right)^{\alpha }&{\text{if }}\alpha {\text{ is a negative integer,}}\end{cases}}}</annotation>

</semantics>

</math>

where γ is the <a href="Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a> (0.5772...).

For large real arguments z ≫ |α2 − ⁠1/4⁠|, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless α is <a href="Half-integer" title="Half-integer">half-integer</a>) because they have <a href="Zero_of_a_function" title="Zero of a function">zeros</a> all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of arg z one can write an equation containing a term of order |z|−1:<a href="#cite_note-41">[41]</a>

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<mrow>

<mrow>

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</mfrac>

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<mfrac>

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<msup>

<mrow>

<mrow>

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</msup>

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<mrow>

<mrow>

</mrow>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{\alpha }(z)&={\sqrt {\frac {2}{\pi z}}}\left(\cos \left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)+e^{\left|\operatorname {Im} (z)\right|}{\mathcal {O}}\left(|z|^{-1}\right)\right)&&{\text{for }}\left|\arg z\right|<\pi ,\\Y_{\alpha }(z)&={\sqrt {\frac {2}{\pi z}}}\left(\sin \left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)+e^{\left|\operatorname {Im} (z)\right|}{\mathcal {O}}\left(|z|^{-1}\right)\right)&&{\text{for }}\left|\arg z\right|<\pi .\end{aligned}}}</annotation>

</semantics>

</math>

(For α = ⁠1/2⁠, the last terms in these formulas drop out completely; see the spherical Bessel functions above.)

The asymptotic forms for the Hankel functions are:

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<mtr>

<mtd>

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<mtd>

<msqrt>

<mfrac>

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</msqrt>

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<msup>

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<mfrac>

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<mtd>

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</mtd>

</mtr>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-\pi <\arg z<2\pi ,\\H_{\alpha }^{(2)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-2\pi <\arg z<\pi .\end{aligned}}}</annotation>

</semantics>

</math>

These can be extended to other values of arg z using equations relating H(1) α(zeimπ) and H(2) α(zeimπ) to H(1) α(z) and H(2) α(z).<a href="#cite_note-42">[42]</a>

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, Jα(z) is not asymptotic to the average of these two asymptotic forms when z is negative (because one or the other will not be correct there, depending on the arg z used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for complex (non-real) z so long as |z| goes to infinity at a constant phase angle arg z (using the square root having positive real part):

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<mtr>

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<mtr>

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</msub>

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<mtd>

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<msup>

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<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{\alpha }(z)&\sim {\frac {1}{\sqrt {2\pi z}}}e^{i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-\pi <\arg z<0,\\[1ex]J_{\alpha }(z)&\sim {\frac {1}{\sqrt {2\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}0<\arg z<\pi ,\\[1ex]Y_{\alpha }(z)&\sim -i{\frac {1}{\sqrt {2\pi z}}}e^{i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-\pi <\arg z<0,\\[1ex]Y_{\alpha }(z)&\sim i{\frac {1}{\sqrt {2\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}0<\arg z<\pi .\end{aligned}}}</annotation>

</semantics>

</math>

For the modified Bessel functions, <a href="Hermann_Hankel" title="Hermann Hankel">Hankel</a> developed <a href="Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansions</a> as well:<a href="#cite_note-43">[43]</a><a href="#cite_note-44">[44]</a>

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<mfrac>

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<mrow>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{\alpha }(z)&\sim {\frac {e^{z}}{\sqrt {2\pi z}}}\left(1-{\frac {4\alpha ^{2}-1}{8z}}+{\frac {\left(4\alpha ^{2}-1\right)\left(4\alpha ^{2}-9\right)}{2!(8z)^{2}}}-{\frac {\left(4\alpha ^{2}-1\right)\left(4\alpha ^{2}-9\right)\left(4\alpha ^{2}-25\right)}{3!(8z)^{3}}}+\cdots \right)&&{\text{for }}\left|\arg z\right|<{\frac {\pi }{2}},\\K_{\alpha }(z)&\sim {\sqrt {\frac {\pi }{2z}}}e^{-z}\left(1+{\frac {4\alpha ^{2}-1}{8z}}+{\frac {\left(4\alpha ^{2}-1\right)\left(4\alpha ^{2}-9\right)}{2!(8z)^{2}}}+{\frac {\left(4\alpha ^{2}-1\right)\left(4\alpha ^{2}-9\right)\left(4\alpha ^{2}-25\right)}{3!(8z)^{3}}}+\cdots \right)&&{\text{for }}\left|\arg z\right|<{\frac {3\pi }{2}}.\end{aligned}}}</annotation>

</semantics>

</math>

There is also the asymptotic form (for large real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

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

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">)<a href="#cite_note-45">[45]</a>

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<mrow>

<mrow>

<mi>arcsinh</mi>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{\alpha }(z)={\frac {1}{{\sqrt {2\pi z}}{\sqrt[{4}]{1+{\frac {\alpha ^{2}}{z^{2}}}}}}}\exp \left(-\alpha \operatorname {arcsinh} \left({\frac {\alpha }{z}}\right)+z{\sqrt {1+{\frac {\alpha ^{2}}{z^{2}}}}}\right)\left(1+{\mathcal {O}}\left({\frac {1}{z{\sqrt {1+{\frac {\alpha ^{2}}{z^{2}}}}}}}\right)\right).\end{aligned}}}</annotation>

</semantics>

</math>

When α = ⁠1/2⁠, all the terms except the first vanish, and we have

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{{1}/{2}}(z)&={\sqrt {\frac {2}{\pi }}}{\frac {\sinh(z)}{\sqrt {z}}}\sim {\frac {e^{z}}{\sqrt {2\pi z}}}&&{\text{for }}\left|\arg z\right|<{\tfrac {\pi }{2}},\\[1ex]K_{{1}/{2}}(z)&={\sqrt {\frac {\pi }{2}}}{\frac {e^{-z}}{\sqrt {z}}}.\end{aligned}}}</annotation>

</semantics>

</math>

For small arguments <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<|z|\ll {\sqrt {\alpha +1}}}">

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<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle 0<|z|\ll {\sqrt {\alpha +1}}}</annotation>

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</math><img src="./0c1df46ae489322dc0d34096b8ab15f1a7086387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.683ex; height:3.009ex;" alt="{\displaystyle 0<|z|\ll {\sqrt {\alpha +1}}}" loading="lazy">, we have

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{\alpha }(z)&\sim {\frac {1}{\Gamma (\alpha +1)}}\left({\frac {z}{2}}\right)^{\alpha },\\[1ex]K_{\alpha }(z)&\sim {\begin{cases}-\ln \left({\dfrac {z}{2}}\right)-\gamma &{\text{if }}\alpha =0\\[1ex]{\frac {\Gamma (\alpha )}{2}}\left({\dfrac {2}{z}}\right)^{\alpha }&{\text{if }}\alpha >0\end{cases}}\end{aligned}}}</annotation>

</semantics>

</math>

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

For integer order α = n, Jn is often defined via a <a href="Laurent_series" title="Laurent series">Laurent series</a> for a generating function:

<msup>

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<mrow>

<mrow>

<mfrac>

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</msup>

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</mrow>

</munderover>

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</msub>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{{\frac {x}{2}}\left(t-{\frac {1}{t}}\right)}=\sum _{n=-\infty }^{\infty }J_{n}(x)t^{n}}</annotation>

</semantics>

</math>

an approach used by <a href="P._A._Hansen" class="mw-redirect" title="P. A. Hansen">P. A. Hansen</a> in 1843. (This can be generalized to non-integer order by <a href="Methods_of_contour_integration" class="mw-redirect" title="Methods of contour integration">contour integration</a> or other methods.)

Infinite series of Bessel functions in the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{\nu =-\infty }^{\infty }J_{N\nu +p}(x)}">

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<annotation encoding="application/x-tex">{\textstyle \sum _{\nu =-\infty }^{\infty }J_{N\nu +p}(x)}</annotation>

</semantics>

</math><img src="./79fd4d3c5bf4a7b0fbf2cf53deb198ac3441a961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.242ex; height:3.176ex;" alt="{\textstyle \sum _{\nu =-\infty }^{\infty }J_{N\nu +p}(x)}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\nu ,p\in \mathbb {Z} ,\ N\in \mathbb {Z} ^{+}">

<mrow>

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<msup>

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<annotation encoding="application/x-tex">\nu ,p\in \mathbb {Z} ,\ N\in \mathbb {Z} ^{+}</annotation>

</semantics>

</math><img src="./972bbac800c102a3dc82c7409c93186d0c1dd80c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.406ex; height:2.843ex;" alt="\nu ,p\in \mathbb {Z} ,\ N\in \mathbb {Z} ^{+}" loading="lazy"> arise in many physical systems and are defined in closed form by the Sung series.<a href="#cite_note-SungSeries-46">[46]</a> For example, when N = 3: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{\nu =-\infty }^{\infty }J_{3\nu +p}(x)={\frac {1}{3}}\left[1+2\cos {(x{\sqrt {3}}/2-2\pi p/3)}\right]}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \sum _{\nu =-\infty }^{\infty }J_{3\nu +p}(x)={\frac {1}{3}}\left[1+2\cos {(x{\sqrt {3}}/2-2\pi p/3)}\right]}</annotation>

</semantics>

</math><img src="./1eaa4c63e9da373d3678e16136bd9a11688985bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:50.13ex; height:3.676ex;" alt="{\textstyle \sum _{\nu =-\infty }^{\infty }J_{3\nu +p}(x)={\frac {1}{3}}\left[1+2\cos {(x{\sqrt {3}}/2-2\pi p/3)}\right]}" loading="lazy">. More generally, the Sung series and the alternating Sung series are written as:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }J_{N\nu +p}(x)={\frac {1}{N}}\sum _{q=0}^{N-1}e^{ix\sin {2\pi q/N}}e^{-i2\pi pq/N}}</annotation>

</semantics>

</math>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }(-1)^{\nu }J_{N\nu +p}(x)={\frac {1}{N}}\sum _{q=0}^{N-1}e^{ix\sin {(2q+1)\pi /N}}e^{-i(2q+1)\pi p/N}}</annotation>

</semantics>

</math>

A series expansion using Bessel functions (<a href="Kapteyn_series" title="Kapteyn series">Kapteyn series</a>) is

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-z}}=1+2\sum _{n=1}^{\infty }J_{n}(nz).}</annotation>

</semantics>

</math>

Another important relation for integer orders is the <a href="Jacobi%E2%80%93Anger_expansion" title="Jacobi–Anger expansion">Jacobi–Anger expansion</a>:

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{iz\cos \phi }=\sum _{n=-\infty }^{\infty }i^{n}J_{n}(z)e^{in\phi }}</annotation>

</semantics>

</math>

and

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\pm iz\sin \phi }=J_{0}(z)+2\sum _{n=1}^{\infty }J_{2n}(z)\cos(2n\phi )\pm 2i\sum _{n=0}^{\infty }J_{2n+1}(z)\sin((2n+1)\phi )}</annotation>

</semantics>

</math>

which is used to expand a <a href="Plane_wave" title="Plane wave">plane wave</a> as a <a href="Plane_wave_expansion" class="mw-redirect" title="Plane wave expansion">sum of cylindrical waves</a>, or to find the <a href="Fourier_series" title="Fourier series">Fourier series</a> of a tone-modulated <a href="Frequency_modulation" title="Frequency modulation">FM</a> signal.

More generally, a series

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=a_{0}^{\nu }J_{\nu }(z)+2\cdot \sum _{k=1}^{\infty }a_{k}^{\nu }J_{\nu +k}(z)}</annotation>

</semantics>

</math>

is called Neumann expansion of f. The coefficients for ν = 0 have the explicit form

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{k}^{0}={\frac {1}{2\pi i}}\int _{|z|=c}f(z)O_{k}(z)\,dz}</annotation>

</semantics>

</math>

where Ok is <a href="Neumann_polynomial" title="Neumann polynomial">Neumann's polynomial</a>.<a href="#cite_note-47">[47]</a>

Selected functions admit the special representation

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}^{\nu }J_{\nu +2k}(z)}</annotation>

</semantics>

</math>

with

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{k}^{\nu }=2(\nu +2k)\int _{0}^{\infty }f(z){\frac {J_{\nu +2k}(z)}{z}}\,dz}</annotation>

</semantics>

</math>

due to the orthogonality relation

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }J_{\alpha }(z)J_{\beta }(z){\frac {dz}{z}}={\frac {2}{\pi }}{\frac {\sin \left({\frac {\pi }{2}}(\alpha -\beta )\right)}{\alpha ^{2}-\beta ^{2}}}}</annotation>

</semantics>

</math>

More generally, if f has a branch-point near the origin of such a nature that

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=\sum _{k=0}a_{k}J_{\nu +k}(z)}</annotation>

</semantics>

</math>

then

</mrow>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mrow>

</munder>

<mfrac>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\left\{\sum _{k=0}a_{k}J_{\nu +k}\right\}(s)={\frac {1}{\sqrt {1+s^{2}}}}\sum _{k=0}{\frac {a_{k}}{\left(s+{\sqrt {1+s^{2}}}\right)^{\nu +k}}}}</annotation>

</semantics>

</math>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}a_{k}\xi ^{\nu +k}={\frac {1+\xi ^{2}}{2\xi }}{\mathcal {L}}\{f\}\left({\frac {1-\xi ^{2}}{2\xi }}\right)}</annotation>

</semantics>

</math>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b2734e92c352d7115a47553097a0e35462f915ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.207ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}\{f\}}" loading="lazy"> is the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> of f.<a href="#cite_note-48">[48]</a>

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{\nu }(z)&={\frac {\left({\frac {z}{2}}\right)^{\nu }}{\Gamma \left(\nu +{\frac {1}{2}}\right){\sqrt {\pi }}}}\int _{-1}^{1}e^{izs}\left(1-s^{2}\right)^{\nu -{\frac {1}{2}}}\,ds\\[5px]&={\frac {2}{{\left({\frac {z}{2}}\right)}^{\nu }\cdot {\sqrt {\pi }}\cdot \Gamma \left({\frac {1}{2}}-\nu \right)}}\int _{1}^{\infty }{\frac {\sin zu}{\left(u^{2}-1\right)^{\nu +{\frac {1}{2}}}}}\,du\end{aligned}}}</annotation>

</semantics>

</math>

where ν > −⁠1/2⁠ and z ∈ C.<a href="#cite_note-Zwillinger_2014-49">[49]</a>

This formula is useful especially when working with <a href="Fourier_transforms" class="mw-redirect" title="Fourier transforms">Fourier transforms</a>.

Because Bessel's equation becomes <a href="Hermitian" class="mw-redirect" title="Hermitian">Hermitian</a> (self-adjoint) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</msubsup>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}xJ_{\alpha }\left(xu_{\alpha ,m}\right)J_{\alpha }\left(xu_{\alpha ,n}\right)\,dx={\frac {\delta _{m,n}}{2}}\left[J_{\alpha +1}\left(u_{\alpha ,m}\right)\right]^{2}={\frac {\delta _{m,n}}{2}}\left[J_{\alpha }'\left(u_{\alpha ,m}\right)\right]^{2}}</annotation>

</semantics>

</math>

where α > −1, δm,n is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, and uα,m is the mth <a href="Root_of_a_function" class="mw-redirect" title="Root of a function">zero</a> of Jα(x). This orthogonality relation can then be used to extract the coefficients in the <a href="Fourier%E2%80%93Bessel_series" title="Fourier–Bessel series">Fourier–Bessel series</a>, where a function is expanded in the basis of the functions Jα(x uα,m) for fixed α and varying m.

An analogous relationship for the spherical Bessel functions follows immediately:

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}x^{2}j_{\alpha }\left(xu_{\alpha ,m}\right)j_{\alpha }\left(xu_{\alpha ,n}\right)\,dx={\frac {\delta _{m,n}}{2}}\left[j_{\alpha +1}\left(u_{\alpha ,m}\right)\right]^{2}}</annotation>

</semantics>

</math>

If one defines a <a href="Boxcar_function" title="Boxcar function">boxcar function</a> of x that depends on a small parameter ε as:

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{\varepsilon }(x)={\frac {1}{\varepsilon }}\operatorname {rect} \left({\frac {x-1}{\varepsilon }}\right)}</annotation>

</semantics>

</math>

(where rect is the <a href="Rectangle_function" class="mw-redirect" title="Rectangle function">rectangle function</a>) then the <a href="Hankel_transform" title="Hankel transform">Hankel transform</a> of it (of any given order α > −⁠1/2⁠), gε(k), approaches Jα(k) as ε approaches zero, for any given k. Conversely, the Hankel transform (of the same order) of gε(k) is fε(x):

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }kJ_{\alpha }(kx)g_{\varepsilon }(k)\,dk=f_{\varepsilon }(x)}</annotation>

</semantics>

</math>

which is zero everywhere except near 1. As ε approaches zero, the right-hand side approaches δ(x − 1), where δ is the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. This admits the limit (in the <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributional</a> sense):

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }kJ_{\alpha }(kx)J_{\alpha }(k)\,dk=\delta (x-1)}</annotation>

</semantics>

</math>

A change of variables then yields the closure equation:<a href="#cite_note-50">[50]</a>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }xJ_{\alpha }(ux)J_{\alpha }(vx)\,dx={\frac {1}{u}}\delta (u-v)}</annotation>

</semantics>

</math>

for α > −⁠1/2⁠. For the spherical Bessel functions the orthogonality relation is:

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }x^{2}j_{\alpha }(ux)j_{\alpha }(vx)\,dx={\frac {\pi }{2uv}}\delta (u-v)}</annotation>

</semantics>

</math>

for α > −1.

Another important property of Bessel's equations, which follows from <a href="Abel's_identity" title="Abel's identity">Abel's identity</a>, involves the <a href="Wronskian" title="Wronskian">Wronskian</a> of the solutions:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

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<mrow>

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</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

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</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{\alpha }(x){\frac {dB_{\alpha }}{dx}}-{\frac {dA_{\alpha }}{dx}}B_{\alpha }(x)={\frac {C_{\alpha }}{x}}}</annotation>

</semantics>

</math>

where Aα and Bα are any two solutions of Bessel's equation, and Cα is a constant independent of x (which depends on α and on the particular Bessel functions considered). In particular,

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

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</msub>

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<mrow>

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</mrow>

<mfrac>

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</msub>

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<msub>

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<mfrac>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }(x){\frac {dY_{\alpha }}{dx}}-{\frac {dJ_{\alpha }}{dx}}Y_{\alpha }(x)={\frac {2}{\pi x}}}</annotation>

</semantics>

</math>

and

<msub>

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</msub>

<mfrac>

<mrow>

<msub>

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<mrow>

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</msub>

<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{\alpha }(x){\frac {dK_{\alpha }}{dx}}-{\frac {dI_{\alpha }}{dx}}K_{\alpha }(x)=-{\frac {1}{x}},}</annotation>

</semantics>

</math>

for α > −1.

For α > −1, the even entire function of genus 1, x−αJα(x), has only real zeros. Let

<msub>

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<annotation encoding="application/x-tex">{\displaystyle 0<j_{\alpha ,1}<j_{\alpha ,2}<\cdots <j_{\alpha ,n}<\cdots }</annotation>

</semantics>

</math>

be all its positive zeros, then

<msub>

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</munderover>

<mrow>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }(z)={\frac {\left({\frac {z}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{j_{\alpha ,n}^{2}}}\right)}</annotation>

</semantics>

</math>

(There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)

<div class="mw-heading mw-heading3"><h3 id="Recurrence_relations">Recurrence relations</h3></div>

The functions Jα, Yα, H(1) α, and H(2) α all satisfy the <a href="Recurrence_relation" title="Recurrence relation">recurrence relations</a><a href="#cite_note-51">[51]</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2\alpha }{x}}Z_{\alpha }(x)=Z_{\alpha -1}(x)+Z_{\alpha +1}(x)}</annotation>

</semantics>

</math>

and

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2{\frac {dZ_{\alpha }(x)}{dx}}=Z_{\alpha -1}(x)-Z_{\alpha +1}(x),}</annotation>

</semantics>

</math>

where Z denotes J, Y, H(1), or H(2). These two identities are often combined, e.g. added or subtracted, to yield various other relations. In this way, for example, one can compute Bessel functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives). In particular, it follows that<a href="#cite_note-52">[52]</a>

<mtr>

<mtd>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{m}\left[x^{\alpha }Z_{\alpha }(x)\right]&=x^{\alpha -m}Z_{\alpha -m}(x),\\\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{m}\left[{\frac {Z_{\alpha }(x)}{x^{\alpha }}}\right]&=(-1)^{m}{\frac {Z_{\alpha +m}(x)}{x^{\alpha +m}}}.\end{aligned}}}</annotation>

</semantics>

</math>

Using the previous relations one can arrive to similar relations for the Spherical Bessel functions:

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2\alpha +1}{x}}j_{\alpha }(x)=j_{\alpha -1}+j_{\alpha +1}}</annotation>

</semantics>

</math>

and

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dj_{\alpha }(x)}{dx}}=j_{\alpha -1}-{\frac {\alpha +1}{x}}j_{\alpha }}</annotation>

</semantics>

</math>

Modified Bessel functions follow similar relations:

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\left({\frac {x}{2}}\right)\left(t+{\frac {1}{t}}\right)}=\sum _{n=-\infty }^{\infty }I_{n}(x)t^{n}}</annotation>

</semantics>

</math>

and

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{z\cos \theta }=I_{0}(z)+2\sum _{n=1}^{\infty }I_{n}(z)\cos n\theta }</annotation>

</semantics>

</math>

and

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }e^{z\cos(m\theta )+y\cos \theta }d\theta =I_{0}(z)I_{0}(y)+2\sum _{n=1}^{\infty }I_{n}(z)I_{mn}(y).}</annotation>

</semantics>

</math>

The recurrence relation reads

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}C_{\alpha -1}(x)-C_{\alpha +1}(x)&={\frac {2\alpha }{x}}C_{\alpha }(x),\\[1ex]C_{\alpha -1}(x)+C_{\alpha +1}(x)&=2{\frac {d}{dx}}C_{\alpha }(x),\end{aligned}}}</annotation>

</semantics>

</math>

where Cα denotes Iα or eαiπKα. These recurrence relations are useful for discrete diffusion problems.

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

In 1929, <a href="Carl_Ludwig_Siegel" title="Carl Ludwig Siegel">Carl Ludwig Siegel</a> proved that Jν(x), J'ν(x), and the <a href="Logarithmic_derivative" title="Logarithmic derivative">logarithmic derivative</a> ⁠J'ν(x)/Jν(x)⁠ are <a href="Transcendental_number" title="Transcendental number">transcendental numbers</a> when ν is rational and x is algebraic and nonzero.<a href="#cite_note-53">[53]</a> The same proof also implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (v+1)(2/x)^{v}J_{v}(x)}">

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (v+1)(2/x)^{v}J_{v}(x)}</annotation>

</semantics>

</math><img src="./e054546eeee39ad6174e153a9398d785aec1eb27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.345ex; height:2.843ex;" alt="{\displaystyle \Gamma (v+1)(2/x)^{v}J_{v}(x)}" loading="lazy"> is transcendental under the same assumptions.<a href="#cite_note-euclid-54">[54]</a>

<div class="mw-heading mw-heading3"><h3 id="Sums_with_Bessel_functions">Sums with Bessel functions</h3></div>

The product of two Bessel functions admits the following sum:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }J_{\nu }(x)J_{n-\nu }(y)=J_{n}(x+y),}</annotation>

</semantics>

</math>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }J_{\nu }(x)J_{\nu +n}(y)=J_{n}(y-x).}</annotation>

</semantics>

</math>

From these equalities it follows that

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }J_{\nu }(x)J_{\nu +n}(x)=\delta _{n,0}}</annotation>

</semantics>

</math>

and as a consequence

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }J_{\nu }^{2}(x)=1.}</annotation>

</semantics>

</math>

These sums can be extended to include a term multiplier that is a polynomial function of the index. For example,

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }\nu J_{\nu }(x)J_{\nu +n}(x)={\frac {x}{2}}\left(\delta _{n,1}+\delta _{n,-1}\right),}</annotation>

</semantics>

</math>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }\nu J_{\nu }^{2}(x)=0,}</annotation>

</semantics>

</math>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }\nu ^{2}J_{\nu }(x)J_{\nu +n}(x)={\frac {x}{2}}\left(\delta _{n,-1}-\delta _{n,1}\right)+{\frac {x^{2}}{4}}\left(\delta _{n,-2}+2\delta _{n,0}+\delta _{n,2}\right),}</annotation>

</semantics>

</math>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\nu =-\infty }^{\infty }\nu ^{2}J_{\nu }^{2}(x)={\frac {x^{2}}{2}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Multiplication_theorem">Multiplication theorem</h2></div>

The Bessel functions obey a <a href="Multiplication_theorem" title="Multiplication theorem">multiplication theorem</a>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda ^{-\nu }J_{\nu }(\lambda z)=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left({\frac {\left(1-\lambda ^{2}\right)z}{2}}\right)^{n}J_{\nu +n}(z),}</annotation>

</semantics>

</math>

where λ and ν may be taken as arbitrary complex numbers.<a href="#cite_note-Abramowitz_9_1_74-55">[55]</a><a href="#cite_note-56">[56]</a> For |λ2 − 1| < 1,<a href="#cite_note-Abramowitz_9_1_74-55">[55]</a> the above expression also holds if J is replaced by Y. The analogous identities for modified Bessel functions and |λ2 − 1| < 1 are

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda ^{-\nu }I_{\nu }(\lambda z)=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left({\frac {\left(\lambda ^{2}-1\right)z}{2}}\right)^{n}I_{\nu +n}(z)}</annotation>

</semantics>

</math>

and

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda ^{-\nu }K_{\nu }(\lambda z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\left({\frac {\left(\lambda ^{2}-1\right)z}{2}}\right)^{n}K_{\nu +n}(z).}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Zeros_of_the_Bessel_function">Zeros of the Bessel function</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Bourget's_hypothesis">Bourget's hypothesis</h3></div>

Bessel himself originally proved that for nonnegative integers n, the equation Jn(x) = 0 has an infinite number of solutions in x.<a href="#cite_note-57">[57]</a> When the functions Jn(x) are plotted on the same graph, though, none of the zeros seem to coincide for different values of n except for the zero at x = 0. This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integers n ≥ 0 and m ≥ 1, the functions Jn(x) and Jn + m(x) have no common zeros other than the one at x = 0. The hypothesis was proved by <a href="Carl_Ludwig_Siegel" title="Carl Ludwig Siegel">Carl Ludwig Siegel</a> in 1929.<a href="#cite_note-58">[58]</a>

<div class="mw-heading mw-heading3"><h3 id="Transcendence_2">Transcendence</h3></div>

Siegel proved in 1929 that when ν is rational, all nonzero roots of Jν(x) and J'ν(x) are <a href="Transcendental_number" title="Transcendental number">transcendental</a>,<a href="#cite_note-lorch-59">[59]</a> as are all the roots of Kν(x).<a href="#cite_note-euclid-54">[54]</a> It is also known that all roots of the higher derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\nu }^{(n)}(x)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1e4aa531d219a6f02042ee99ace5c1a84181f208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.163ex; height:3.509ex;" alt="{\displaystyle J_{\nu }^{(n)}(x)}" loading="lazy"> for n ≤ 18 are transcendental, except for the special values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{1}^{(3)}(\pm {\sqrt {3}})=0}">

</mrow>

</mrow>

</msubsup>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{1}^{(3)}(\pm {\sqrt {3}})=0}</annotation>

</semantics>

</math><img src="./75f6feb29d021de65cbb7b5ef8cedaa82804958d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.836ex; height:3.676ex;" alt="{\displaystyle J_{1}^{(3)}(\pm {\sqrt {3}})=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{0}^{(4)}(\pm {\sqrt {3}})=0}">

</mrow>

</mrow>

</msubsup>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{0}^{(4)}(\pm {\sqrt {3}})=0}</annotation>

</semantics>

</math><img src="./c1a8402f813d48dffa592bf2274b4fad07a2a283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.836ex; height:3.676ex;" alt="{\displaystyle J_{0}^{(4)}(\pm {\sqrt {3}})=0}" loading="lazy">.<a href="#cite_note-lorch-59">[59]</a>

<div class="mw-heading mw-heading3"><h3 id="Numerical_approaches">Numerical approaches</h3></div>

For numerical studies about the zeros of the Bessel function, see <a href="#CITEREFGilSeguraTemme2007">Gil, Segura & Temme (2007)</a>, <a href="#CITEREFKravanjaRagosVrahatisZafiropoulos1998">Kravanja et al. (1998)</a> and <a href="#CITEREFMoler2004">Moler (2004)</a>.

<div class="mw-heading mw-heading3"><h3 id="Numerical_values">Numerical values</h3></div>

The first zeros in J0 (i.e., j0,1, j0,2 and j0,3) occur at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.<a href="#cite_note-60">[60]</a>

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

<div class="mw-heading mw-heading3"><h3 id="Waves_and_elasticity_problems">Waves and elasticity problems</h3></div>

The first appearance of a Bessel function appears in the work of <a href="Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> in 1732, while working on the analysis of a <a href="String_vibration" title="String vibration">vibrating string</a>, a problem that was tackled before by his father <a href="Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a>.<a href="#cite_note-:0-1">[1]</a> Daniel considered a flexible chain suspended from a fixed point above and free at its lower end.<a href="#cite_note-:0-1">[1]</a> The solution of the differential equation led to the introduction of a function that is now considered <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{0}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

<a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in 1736, found a link between other functions (now known as <a href="Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a>) and Bernoulli's solution. Euler also introduced a non-uniform chain that lead to the introduction of functions now related to modified Bessel functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ddf898aaf9cf058f8b41028bc9ca3c5ac08a9b6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.381ex; height:2.843ex;" alt="{\displaystyle I_{n}(x)}" loading="lazy">.<a href="#cite_note-:0-1">[1]</a>

In the middle of the eighteen century, <a href="Jean_le_Rond_d'Alembert" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a> had found a <a href="D'Alembert's_formula" title="D'Alembert's formula">formula</a> to solve the <a href="Wave_equation" title="Wave equation">wave equation</a>. By 1771 there was dispute between Bernoulli, Euler, d'Alembert and <a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> on the nature of the solutions of vibrating strings.<a href="#cite_note-:0-1">[1]</a>

Euler worked in 1778 on <a href="Buckling" title="Buckling">buckling</a>, introducing the concept of <a href="Euler's_critical_load" title="Euler's critical load">Euler's critical load</a>. To solve the problem he introduced the series for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\pm 1/3}(x)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{\pm 1/3}(x)}</annotation>

</semantics>

</math><img src="./c74554bcc75af1fef3bd6aff9fdd294098cb861f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.406ex; height:3.176ex;" alt="{\displaystyle J_{\pm 1/3}(x)}" loading="lazy">.<a href="#cite_note-:0-1">[1]</a> Euler also worked out the solutions of vibrating 2D membranes in cylindrical coordinates in 1780. In order to solve his differential equation he introduced a power series associated to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3462eec70865e8698d6ae2f4a576a3bb949661e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.648ex; height:2.843ex;" alt="{\displaystyle J_{n}(x)}" loading="lazy">, for integer n.<a href="#cite_note-:0-1">[1]</a>

During the end of the 19th century Lagrange, <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> and <a href="Marc-Antoine_Parseval" title="Marc-Antoine Parseval">Marc-Antoine Parseval</a> also found equivalents to the Bessel functions.<a href="#cite_note-:0-1">[1]</a> Parseval for example found an integral representation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{0}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

At the beginning of the 1800s, <a href="Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> used <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{0}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47cd840d13c953bbfc6368b978e6cfc263d74b62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.483ex; height:2.843ex;" alt="{\displaystyle J_{0}(x)}" loading="lazy"> to solve the <a href="Heat_equation" title="Heat equation">heat equation</a> in a problem with cylindrical symmetry.<a href="#cite_note-:0-1">[1]</a> Fourier won a prize of the <a href="French_Academy_of_Sciences" title="French Academy of Sciences">French Academy of Sciences</a> for this work in 1811.<a href="#cite_note-:0-1">[1]</a> But most of the details of his work, including the use of a <a href="Fourier_series" title="Fourier series">Fourier series</a>, remained unpublished until 1822.<a href="#cite_note-:0-1">[1]</a> Poisson in rivalry with Fourier, extended Fourier's work in 1823, introducing new properties of Bessel functions including Bessel functions of half-integer order (now known as spherical Bessel functions).<a href="#cite_note-:0-1">[1]</a>

<div class="mw-heading mw-heading3"><h3 id="Astronomical_problems">Astronomical problems</h3></div>

In 1770, Lagrange introduced the series expansion of Bessel functions to solve <a href="Kepler's_equation" title="Kepler's equation">Kepler's equation</a>, a transcendental equation in astronomy. <a href="Friedrich_Wilhelm_Bessel" title="Friedrich Wilhelm Bessel">Friedrich Wilhelm Bessel</a> had seen Lagrange's solution but found it difficult to handle. In 1813 in a letter to <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>, Bessel simplified the calculation using trigonometric functions.<a href="#cite_note-:0-1">[1]</a> Bessel published his work in 1819, independently introducing the method of Fourier series unaware of the work of Fourier which was published later.<a href="#cite_note-:0-1">[1]</a>

In 1824, Bessel carried out a systematic investigation of the functions, which earned the functions his name.<a href="#cite_note-:0-1">[1]</a> In older literature the functions were called cylindrical functions or even Bessel–Fourier functions.<a href="#cite_note-:0-1">[1]</a>

/* start https://en.wikipedia.org/ */

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<ul><li><a href="Anger_function" title="Anger function">Anger function</a></li>

<li><a href="Bessel_polynomials" title="Bessel polynomials">Bessel polynomials</a></li>

<li><a href="Bessel%E2%80%93Clifford_function" title="Bessel–Clifford function">Bessel–Clifford function</a></li>

<li><a href="Bessel%E2%80%93Maitland_function" title="Bessel–Maitland function">Bessel–Maitland function</a></li>

<li><a href="Fourier%E2%80%93Bessel_series" title="Fourier–Bessel series">Fourier–Bessel series</a></li>

<li><a href="Hahn%E2%80%93Exton_q-Bessel_function" title="Hahn–Exton q-Bessel function">Hahn–Exton q-Bessel function</a></li>

<li><a href="Hankel_transform" title="Hankel transform">Hankel transform</a></li>

<li><a href="Incomplete_Bessel_functions" title="Incomplete Bessel functions">Incomplete Bessel functions</a></li>

<li><a href="Jackson_q-Bessel_function" title="Jackson q-Bessel function">Jackson q-Bessel function</a></li>

<li><a href="Kelvin_functions" title="Kelvin functions">Kelvin functions</a></li>

<li><a href="Kontorovich%E2%80%93Lebedev_transform" title="Kontorovich–Lebedev transform">Kontorovich–Lebedev transform</a></li>

<li><a href="Lentz's_algorithm" title="Lentz's algorithm">Lentz's algorithm</a></li>

<li><a href="Lerche%E2%80%93Newberger_sum_rule" title="Lerche–Newberger sum rule">Lerche–Newberger sum rule</a></li>

<li><a href="Lommel_function" title="Lommel function">Lommel function</a></li>

<li><a href="Lommel_polynomial" title="Lommel polynomial">Lommel polynomial</a></li>

<li><a href="Neumann_polynomial" title="Neumann polynomial">Neumann polynomial</a></li>

<li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Riccati-BesselFunctions.html">Riccati-Bessel Functions</a></li>

<li><a href="Schl%C3%B6milch's_series" title="Schlömilch's series">Schlömilch's series</a></li>

<li><a href="Sonine_formula" title="Sonine formula">Sonine formula</a></li>

<li><a href="Struve_function" title="Struve function">Struve function</a></li>

<li><a href="Vibrations_of_a_circular_membrane" class="mw-redirect" title="Vibrations of a circular membrane">Vibrations of a circular membrane</a></li>

<li>Weber function (defined at <a href="Anger_function" title="Anger function">Anger function</a>)</li>

<li><a href="Gauss_circle_problem#Exact_forms" title="Gauss circle problem">Gauss' circle problem</a></li></ul>

</div>

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

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</style><cite id="CITEREFDutka1995" class="citation journal cs1">Dutka, Jacques (1995). "On the early history of Bessel functions". Archive for History of Exact Sciences. 49 (2): 105–134. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00376544">10.1007/BF00376544</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWilenskyBrownHazelton2023" class="citation journal cs1">Wilensky, Michael; Brown, Jordan; Hazelton, Bryna (June 2023). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2Fstad863">"Why and when to expect Gaussian error distributions in epoch of reionization 21-cm power spectrum measurements"</a>. Monthly Notices of the Royal Astronomical Society. 521 (4): 5191–5206. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2211.13576">2211.13576</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmnras%2Fstad863">10.1093/mnras/stad863</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <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/SphericalBesselFunctionoftheSecondKind.html">"Spherical Bessel Function of the Second Kind"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-MathWorld-4">^ <a href="#cite_ref-MathWorld_4-0">a</a> <a href="#cite_ref-MathWorld_4-1">b</a> <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/BesselFunctionoftheSecondKind.html">"Bessel Function of the Second Kind"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-p360-5">^ <a href="#cite_ref-p360_5-0">a</a> <a href="#cite_ref-p360_5-1">b</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_360.htm">p. 360, 9.1.10</a>.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFWhittakerWatson1927" class="citation book cs1"><a href="Edmund_T._Whittaker" class="mw-redirect" title="Edmund T. Whittaker">Whittaker, Edmund Taylor</a>; <a href="George_N._Watson" class="mw-redirect" title="George N. Watson">Watson, George Neville</a> (1927). <a href="A_Course_of_Modern_Analysis" title="A Course of Modern Analysis">A Course of Modern Analysis</a> (4th ed.). Cambridge University Press. p. 356.</cite> For example, Hansen (1843) and Schlömilch (1857).

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_358.htm">p. 358, 9.1.5</a>.

</li>

<li id="cite_note-Temme-8">^ <a href="#cite_ref-Temme_8-0">a</a> <a href="#cite_ref-Temme_8-1">b</a> <cite id="CITEREFTemme1996" class="citation book cs1">Temme, Nico M. (1996). Special Functions: An introduction to the classical functions of mathematical physics (2nd print ed.). New York: Wiley. pp. 228–231. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0471113131</bdi>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <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/Hansen-BesselFormula.html">"Hansen-Bessel Formula"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Bessel, F. (1824). The relevant integral is an unnumbered equation between equations 28 and 29. Note that Bessel's <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{k}^{h}}">

</mrow>

</mrow>

</msubsup>

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

</semantics>

</math><img src="./92f261c5724928fb5b6f9fd740a451b8a9673454.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.396ex; height:3.176ex;" alt="{\displaystyle I_{k}^{h}}" loading="lazy"> would today be written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{h}(k)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6826f14cb421f6dde9bb5fbb1a85d20548e510ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.49ex; height:2.843ex;" alt="{\displaystyle J_{h}(k)}" loading="lazy">.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> Watson, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA176">p. 176</a>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100923194031/http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html">"Properties of Hankel and Bessel Functions"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html">the original</a> on 2010-09-23. Retrieved 2010-10-18.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20221003054117/https://www.nbi.dk/~polesen/borel/node15.html">"Integral representations of the Bessel function"</a>. www.nbi.dk. Archived from <a rel="nofollow" class="external text" href="https://www.nbi.dk/~polesen/borel/node15.html">the original</a> on 3 October 2022. Retrieved 25 March 2018.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> Arfken & Weber, exercise 11.1.17.

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_362.htm">p. 362, 9.1.69</a>.

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFSzegő1975" class="citation book cs1"><a href="G%C3%A1bor_Szeg%C5%91" title="Gábor Szegő">Szegő, Gábor</a> (1975). Orthogonal Polynomials (4th ed.). Providence, RI: AMS.</cite>

</li>

<li id="cite_note-mhtlab.uwaterloo.ca-17"><a href="#cite_ref-mhtlab.uwaterloo.ca_17-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf">"Bessel Functions of the First and Second Kind"</a> (PDF). mhtlab.uwaterloo.ca. p. 3. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf">Archived</a> (PDF) from the original on 2022-10-09. Retrieved 24 May 2022.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a rel="nofollow" class="external text" href="https://dlmf.nist.gov/10.8#E1">NIST Digital Library of Mathematical Functions</a>, (10.8.1). Accessed on line Oct. 25, 2016.

</li>

<li id="cite_note-p._178-19">^ <a href="#cite_ref-p._178_19-0">a</a> <a href="#cite_ref-p._178_19-1">b</a> Watson, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA178">p. 178</a>.

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_358.htm">p. 358, 9.1.3, 9.1.4</a>.

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_358.htm">p. 358, 9.1.6</a>.

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_360.htm">p. 360, 9.1.25</a>.

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_375.htm">p. 375, 9.6.2, 9.6.10, 9.6.11</a>.

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFDixonFerrar1930" class="citation journal cs1">Dixon; Ferrar, W.L. (1930). "A direct proof of Nicholson's integral". The Quarterly Journal of Mathematics. Oxford: 236–238. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fqmath%2Fos-1.1.236">10.1093/qmath/os-1.1.236</a>.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_375.htm">p. 375, 9.6.3, 9.6.5</a>.

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_374.htm">p. 374, 9.6.1</a>.

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFGreinerReinhardt2009" class="citation book cs1">Greiner, Walter; Reinhardt, Joachim (2009). Quantum Electrodynamics. Springer. p. 72. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-87561-1</bdi>.</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> Watson, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA181">p. 181</a>.

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://dlmf.nist.gov/10.32.E11">"Modified Bessel Functions §10.32 Integral Representations"</a>. NIST Digital Library of Mathematical Functions. NIST. Retrieved 2024-11-20.</cite>

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> <cite id="CITEREFKhokonov2004" class="citation journal cs1">Khokonov, M. Kh. (2004). "Cascade Processes of Energy Loss by Emission of Hard Photons". Journal of Experimental and Theoretical Physics. 99 (4): 690–707. <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/2004JETP...99..690K">2004JETP...99..690K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2F1.1826160">10.1134/1.1826160</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:122599440">122599440</a>.</cite>. Derived from formulas sourced to <a href="Gradshteyn_and_Ryzhik" title="Gradshteyn and Ryzhik">I. S. Gradshteyn and I. M. Ryzhik</a>, <a href="Table_of_Integrals%2C_Series%2C_and_Products" class="mw-redirect" title="Table of Integrals, Series, and Products">Table of Integrals, Series, and Products</a> (Fizmatgiz, Moscow, 1963; Academic Press, New York, 1980).

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> Referred to as such in: <cite id="CITEREFTeichroew1957" class="citation journal cs1">Teichroew, D. (1957). <a rel="nofollow" class="external text" href="https://dml.cz/bitstream/handle/10338.dmlcz/103973/AplMat_27-1982-4_7.pdf">"The Mixture of Normal Distributions with Different Variances"</a> (PDF). The Annals of Mathematical Statistics. 28 (2): 510–512. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177706981">10.1214/aoms/1177706981</a>.</cite>

</li>

<li id="cite_note-32"><a href="#cite_ref-32">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_437.htm">p. 437, 10.1.1</a>.

</li>

<li id="cite_note-33"><a href="#cite_ref-33">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_439.htm">p. 439, 10.1.25, 10.1.26</a>.

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_438.htm">p. 438, 10.1.11</a>.

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_438.htm">p. 438, 10.1.12</a>.

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_439.htm">p. 439, 10.1.39</a>.

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> L.V. Babushkina, M.K. Kerimov, A.I. Nikitin, Algorithms for computing Bessel functions of half-integer order with complex arguments, <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/0041555388900183">p. 110, p. 111</a>.

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_439.htm">p. 439, 10.1.23, 10.1.24</a>.

</li>

<li id="cite_note-39"><a href="#cite_ref-39">^</a> Griffiths. Introduction to Quantum Mechanics, 2nd edition, p. 154.

</li>

<li id="cite_note-40"><a href="#cite_ref-40">^</a> <cite id="CITEREFDu2004" class="citation journal cs1">Du, Hong (2004). "Mie-scattering calculation". Applied Optics. 43 (9): 1951–1956. <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/2004ApOpt..43.1951D">2004ApOpt..43.1951D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2Fao.43.001951">10.1364/ao.43.001951</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15065726">15065726</a>.</cite>

</li>

<li id="cite_note-41"><a href="#cite_ref-41">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_364.htm">p. 364, 9.2.1</a>.

</li>

<li id="cite_note-42"><a href="#cite_ref-42">^</a> <a href="NIST" class="mw-redirect" title="NIST">NIST</a> <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">Digital Library of Mathematical Functions</a>, Section <a rel="nofollow" class="external text" href="https://dlmf.nist.gov/10.11#E1">10.11</a>.

</li>

<li id="cite_note-43"><a href="#cite_ref-43">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_377.htm">p. 377, 9.7.1</a>.

</li>

<li id="cite_note-44"><a href="#cite_ref-44">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_378.htm">p. 378, 9.7.2</a>.

</li>

<li id="cite_note-45"><a href="#cite_ref-45">^</a> <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-81/issue-4/The-Kosterlitz-Thouless-transition-in-two-dimensional-abelian-spin-systems/cmp/1103920388.full">Fröhlich and Spencer 1981 Appendix B</a>

</li>

<li id="cite_note-SungSeries-46"><a href="#cite_ref-SungSeries_46-0">^</a> <cite id="CITEREFSungHovden2022" class="citation arxiv cs1">Sung, S.; Hovden, R. (2022). "On Infinite Series of Bessel functions of the First Kind". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2211.01148">2211.01148</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math-ph">math-ph</a>].</cite>

</li>

<li id="cite_note-47"><a href="#cite_ref-47">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_363.htm">p. 363, 9.1.82</a> ff.

</li>

<li id="cite_note-48"><a href="#cite_ref-48">^</a> <cite id="CITEREFWatson1995" class="citation book cs1">Watson, G. N. (25 August 1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Mlk3FrNoEVoC&q=bessel+neumann+series&pg=PA536">A Treatise on the Theory of Bessel Functions</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521483919</bdi>. Retrieved 25 March 2018 – via Google Books.</cite>

</li>

<li id="cite_note-Zwillinger_2014-49"><a href="#cite_ref-Zwillinger_2014_49-0">^</a> <cite id="CITEREFGradshteynRyzhikGeronimusTseytlin2015" class="citation book cs1"><a href="Izrail_Solomonovich_Gradshteyn" class="mw-redirect" title="Izrail Solomonovich Gradshteyn">Gradshteyn, Izrail Solomonovich</a>; <a href="Iosif_Moiseevich_Ryzhik" class="mw-redirect" title="Iosif Moiseevich Ryzhik">Ryzhik, Iosif Moiseevich</a>; <a href="Yuri_Veniaminovich_Geronimus" class="mw-redirect" title="Yuri Veniaminovich Geronimus">Geronimus, Yuri Veniaminovich</a>; <a href="Michail_Yulyevich_Tseytlin" class="mw-redirect" title="Michail Yulyevich Tseytlin">Tseytlin, Michail Yulyevich</a>; Jeffrey, Alan (2015) [October 2014]. "8.411.10.". In Zwillinger, Daniel; <a href="Victor_Hugo_Moll" class="mw-redirect" title="Victor Hugo Moll">Moll, Victor Hugo</a> (eds.). <a href="Gradshteyn_and_Ryzhik" title="Gradshteyn and Ryzhik">Table of Integrals, Series, and Products</a>. Translated by Scripta Technica, Inc. (8 ed.). <a href="Academic_Press%2C_Inc." class="mw-redirect" title="Academic Press, Inc.">Academic Press, Inc.</a> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-384933-5</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2014010276">2014010276</a>.</cite>

</li>

<li id="cite_note-50"><a href="#cite_ref-50">^</a> Arfken & Weber, section 11.2

</li>

<li id="cite_note-51"><a href="#cite_ref-51">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_361.htm">p. 361, 9.1.27</a>.

</li>

<li id="cite_note-52"><a href="#cite_ref-52">^</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_361.htm">p. 361, 9.1.30</a>.

</li>

<li id="cite_note-53"><a href="#cite_ref-53">^</a> <cite id="CITEREFSiegel2014" class="citation book cs1 cs1-prop-foreign-lang-source">Siegel, Carl L. (2014). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-88-7642-520-2_2">"Über einige Anwendungen diophantischer Approximationen"</a>. On Some Applications of Diophantine Approximations: a translation of Carl Ludwig Siegel's Über einige Anwendungen diophantischer Approximationen by Clemens Fuchs, with a commentary and the article Integral points on curves: Siegel's theorem after Siegel's proof by Clemens Fuchs and Umberto Zannier (in German). Scuola Normale Superiore. pp. 81–138. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-88-7642-520-2_2">10.1007/978-88-7642-520-2_2</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-88-7642-520-2</bdi>.</cite>

</li>

<li id="cite_note-euclid-54">^ <a href="#cite_ref-euclid_54-0">a</a> <a href="#cite_ref-euclid_54-1">b</a> <cite id="CITEREFJames1950" class="citation journal cs1">James, R. D. (November 1950). <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-56/issue-6/Review-Carl-Ludwig-Siegel-Transcendental-numbers/bams/1183515049.full">"Review: Carl Ludwig Siegel, Transcendental numbers"</a>. Bulletin of the American Mathematical Society. 56 (6): 523–526. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1950-09435-X">10.1090/S0002-9904-1950-09435-X</a>.</cite>

</li>

<li id="cite_note-Abramowitz_9_1_74-55">^ <a href="#cite_ref-Abramowitz_9_1_74_55-0">a</a> <a href="#cite_ref-Abramowitz_9_1_74_55-1">b</a> Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_363.htm">p. 363, 9.1.74</a>.

</li>

<li id="cite_note-56"><a href="#cite_ref-56">^</a> <cite id="CITEREFTruesdell1950" class="citation journal cs1">Truesdell, C. (1950). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063284">"On the Addition and Multiplication Theorems for the Special Functions"</a>. Proceedings of the National Academy of Sciences. 1950 (12): 752–757. <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/1950PNAS...36..752T">1950PNAS...36..752T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.36.12.752">10.1073/pnas.36.12.752</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063284">1063284</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578355">16578355</a>.</cite>

</li>

<li id="cite_note-57"><a href="#cite_ref-57">^</a> Bessel, F. (1824), article 14.

</li>

<li id="cite_note-58"><a href="#cite_ref-58">^</a> Watson, pp. 484–485.

</li>

<li id="cite_note-lorch-59">^ <a href="#cite_ref-lorch_59-0">a</a> <a href="#cite_ref-lorch_59-1">b</a> <cite id="CITEREFLorchMuldoon1995" class="citation journal cs1">Lorch, Lee; Muldoon, Martin E. (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1155%2FS0161171295000706">"Transcendentality of zeros of higher dereivatives of functions involving Bessel functions"</a>. International Journal of Mathematics and Mathematical Sciences. 18 (3): 551–560. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1155%2FS0161171295000706">10.1155/S0161171295000706</a>.</cite>

</li>

<li id="cite_note-60"><a href="#cite_ref-60">^</a> Abramowitz & Stegun, p409

</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">

<ul><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_355.htm">"Chapter 9"</a>. <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 355, 435. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <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> <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> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite> See also <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_435.htm">chapter 10</a>.</li>

<li><a href="George_B._Arfken" title="George B. Arfken">Arfken, George B.</a> and Hans J. Weber, Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, 2005). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-059876-0</bdi>.</li>

<li><cite id="CITEREFBessel1824" class="citation journal cs1">Bessel, Friedrich (1824). "Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne entsteht" [Investigation of the part of the planetary disturbances which arise from the movement of the sun]. Berlin Abhandlungen.</cite> Reproduced as pages 84 to 109 in <cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://play.google.com/store/books/details?id=Un4EAAAAYAAJ">Abhandlungen von Friedrich Wilhelm Bessel</a>. Leipzig: Engelmann. 1875.</cite> <a rel="nofollow" class="external text" href="https://drive.google.com/file/d/1W-z4BNN4s7nfGC9IbCkhQksHzGQG-wox/view?usp=sharing">English translation of the text</a>.</li>

<li>Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-60462-4</bdi>.</li>

<li><cite id="CITEREFGilSeguraTemme2007" class="citation book cs1">Gil, A.; Segura, J.; Temme, N. M. (2007). Numerical methods for special functions. Society for Industrial and Applied Mathematics.</cite></li>

<li><cite id="CITEREFKravanjaRagosVrahatisZafiropoulos1998" class="citation cs2">Kravanja, P.; Ragos, O.; Vrahatis, M.N.; Zafiropoulos, F.A. (1998), "ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument", Computer Physics Communications, 113 (2–3): 220–238, <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/1998CoPhC.113..220K">1998CoPhC.113..220K</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%2FS0010-4655%2898%2900064-2">10.1016/S0010-4655(98)00064-2</a></cite></li>

<li><cite id="CITEREFMie1908" class="citation journal cs1"><a href="Gustav_Mie" title="Gustav Mie">Mie, G.</a> (1908). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19083300302">"Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen"</a>. Annalen der Physik. 25 (3): 377. <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/1908AnP...330..377M">1908AnP...330..377M</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.19083300302">10.1002/andp.19083300302</a>.</cite></li>

<li><cite id="CITEREFOlverMaximon2010" class="citation cs2"><a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, F. W. J.</a>; Maximon, L. C. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/10">"Bessel 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.), <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="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2"><a href="William_H._Press" title="William H. Press">Press, W. H.</a>; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210203001225/http://numerical.recipes/book/book.html">"Section 6.5. Bessel Functions of Integer Order"</a>, Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi>, archived from <a rel="nofollow" class="external text" href="http://numerical.recipes/book/book.html">the original</a> on 2021-02-03, retrieved 2022-09-28</cite>.</li>

<li>B Spain, M. G. Smith, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kYgZAQAAIAAJ&q=Bessel">Functions of mathematical physics</a>, Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions.</li>

<li>N. M. Temme, Special Functions. An Introduction to the Classical Functions of Mathematical Physics, John Wiley and Sons, Inc., New York, 1996. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-11313-1</bdi>. Chapter 9 deals with Bessel functions.</li>

<li><a href="G._N._Watson" title="G. N. Watson">Watson, G. N.</a>, A Treatise on the Theory of Bessel Functions, Second Edition, (1995) Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-48391-3</bdi>.</li>

<li><cite id="CITEREFWeber1873" class="citation cs2"><a href="Heinrich_Friedrich_Weber" title="Heinrich Friedrich Weber">Weber, Heinrich</a> (1873), "Ueber eine Darstellung willkürlicher Functionen durch Bessel'sche Functionen", Mathematische Annalen, 6 (2): 146–161, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01443190">10.1007/BF01443190</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:122409461">122409461</a></cite>.</li></ul>

</div>

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

<ul><li><cite id="CITEREFLizorkin2001" class="citation cs2">Lizorkin, P. I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bessel_functions">"Bessel 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></cite>.</li>

<li><cite id="CITEREFKarmazinaPrudnikov2001" class="citation cs2">Karmazina, L. N.; Prudnikov, A.P. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Cylinder_function">"Cylinder function"</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></cite>.</li>

<li><cite id="CITEREFRozov2001" class="citation cs2">Rozov, N. Kh. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bessel_equation">"Bessel equation"</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></cite>.</li>

<li>Wolfram function pages on Bessel <a rel="nofollow" class="external text" href="https://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/">J</a> and <a rel="nofollow" class="external text" href="https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/">Y</a> functions, and modified Bessel <a rel="nofollow" class="external text" href="https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/">I</a> and <a rel="nofollow" class="external text" href="https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/">K</a> functions. Pages include formulas, function evaluators, and plotting calculators.</li>

<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/BesselFunctionoftheFirstKind.html">"Bessel functions of the first kind"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li>Bessel functions <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11/appendix-a-34">Jν</a>, <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11/appendix-a-35">Yν</a>, <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11/appendix-a-36">Iν</a> and <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11/appendix-a-37">Kν</a> in Librow <a rel="nofollow" class="external text" href="http://www.librow.com/articles/article-11">Function handbook</a>.</li>

<li>F. W. J. Olver, L. C. Maximon, <a rel="nofollow" class="external text" href="https://dlmf.nist.gov/10">Bessel Functions</a> (chapter 10 of the Digital Library of Mathematical Functions).</li>

<li><cite id="CITEREFMoler2004" class="citation book cs1">Moler, C. B. (2004). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170808214249/http://tocs.ulb.tu-darmstadt.de/124154883.pdf">Numerical Computing with MATLAB</a> (PDF). Society for Industrial and Applied Mathematics. Archived from <a rel="nofollow" class="external text" href="http://tocs.ulb.tu-darmstadt.de/124154883.pdf">the original</a> (PDF) on 2017-08-08.</cite></li></ul>

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