Editing Bessel function

{{Short description|Families of solutions to related differential equations}}
{{Use American English|date=January 2019}}
{{technical|date=May 2025}}

[[File:Vibrating drum Bessel function.gif|thumb|Bessel functions describe the radial part of [[vibrations of a circular membrane]].]]

'''Bessel functions''', named after [[Friedrich Bessel]] who was the first to systematically study them in 1824,<ref name=":0">{{cite journal |last1=Dutka |first1=Jacques |title=On the early history of Bessel functions |journal=Archive for History of Exact Sciences |date=1995 |volume=49 |issue=2 |pages=105–134 |doi=10.1007/BF00376544}}</ref> are canonical solutions {{math|''y''(''x'')}} of Bessel's [[differential equation]]
<math display="block">x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0</math>
for an arbitrary [[complex number]] <math>\alpha</math>, which represents the ''order'' of the Bessel function. Although <math>\alpha</math> and <math>-\alpha</math> produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly [[smooth function]]s of <math>\alpha</math>.

The most important cases are when <math>\alpha</math> is an [[integer]] or [[half-integer]]. Bessel functions for integer <math>\alpha</math> are also known as '''cylinder functions''' or the '''[[cylindrical harmonics]]''' because they appear in the solution to [[Laplace's equation]] in [[cylindrical coordinates]]. '''[[#Spherical Bessel functions|Spherical Bessel functions]]''' with half-integer <math>\alpha</math> are obtained when solving the [[Helmholtz equation]] in [[spherical coordinates]].

== Applications ==
Bessel's equation arises when finding separable solutions to [[Laplace's equation]] and the [[Helmholtz equation]] in cylindrical or [[spherical coordinates]]. Bessel functions are therefore especially important for many problems of [[wave propagation]] and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order ({{math|1=''α'' = ''n''}}); in spherical problems, one obtains half-integer orders ({{math|1=''α'' = ''n'' + 1/2}}). For example:
* [[Electromagnetic radiation|Electromagnetic waves]] in a cylindrical [[waveguide]]
* Pressure amplitudes of [[inviscid flow|inviscid]] rotational flows
* [[Conduction (heat)|Heat conduction]] in a cylindrical object
* Modes of vibration of a thin circular or annular [[acoustic membrane]] (such as a [[drumhead]] or other [[membranophone]]) or thicker plates such as sheet metal (see [[Kirchhoff–Love plate theory]], [[Mindlin–Reissner plate theory]])
* Diffusion problems on a lattice
* Solutions to the [[Schrödinger equation]] in spherical and cylindrical coordinates for a free particle
* Position space representation of the Feynman [[propagator]] in [[quantum field theory]]
* Solving for patterns of acoustical radiation
* Frequency-dependent friction in circular pipelines
* Dynamics of floating bodies
* [[Angular resolution]]
* Diffraction from helical objects, including [[DNA]]
* [[Probability density function]] of product of two normally distributed random variables<ref>{{cite journal |last1=Wilensky |first1=Michael |last2=Brown |first2=Jordan |last3=Hazelton |first3=Bryna |title=Why and when to expect Gaussian error distributions in epoch of reionization 21-cm power spectrum measurements |journal=Monthly Notices of the Royal Astronomical Society |date=June 2023 |volume=521 |issue=4 |pages=5191–5206 |doi=10.1093/mnras/stad863|doi-access=free |arxiv=2211.13576 }}</ref>
* Analyzing of the surface waves generated by microtremors, in [[geophysics]] and [[seismology]].

Bessel functions also appear in other problems, such as signal processing (e.g., see [[frequency modulation synthesis|FM audio synthesis]], [[Kaiser window]], or [[Bessel filter]]).

== Definitions ==
Because this is a linear differential equation, solutions can be scaled to any amplitude.  The amplitudes chosen for the functions originate from the early work in which the functions appeared as [[#Bessel's_integrals|solutions to definite integrals]] rather than solutions to differential equations.  Because the differential equation is second-order, there must be two [[linear independence|linearly independent]] 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>\alpha</math> when <math>\alpha</math> is known to be an integer.

{| class="wikitable"
! Type !! First kind !! Second kind
|-
| Bessel functions
| {{mvar|[[#Bessel functions of the first kind|J<sub>α</sub>]]}}
| {{mvar|[[#Bessel functions of the second kind|Y<sub>α</sub>]]}}
|-
| [[#Modified Bessel functions|Modified Bessel functions]]
| {{mvar|I<sub>α</sub>}}
| {{mvar|K<sub>α</sub>}}
|-
| [[#Hankel functions|Hankel functions]]
| {{math|1=''H''{{su|b=''α''|p=(1)}} = ''J<sub>α</sub>'' + ''iY<sub>α</sub>''}}
| {{math|1=''H''{{su|b=''α''|p=(2)}} = ''J<sub>α</sub>'' − ''iY<sub>α</sub>''}}
|-
| [[#Spherical Bessel functions|Spherical Bessel functions]]
| {{mvar|j<sub>n</sub>}}
| {{mvar|y<sub>n</sub>}}
|-
| [[#Modified Spherical Bessel functions|Modified spherical Bessel functions]]
| {{mvar|i<sub>n</sub>}}
| {{mvar|k<sub>n</sub>}}
|-
| [[#Spherical Hankel functions|Spherical Hankel functions]]
| {{math|1=''h''{{su|b=''n''|p=(1)}} = ''j<sub>n</sub>'' + ''iy<sub>n</sub>''}}
| {{math|1=''h''{{su|b=''n''|p=(2)}} = ''j<sub>n</sub>'' − ''iy<sub>n</sub>''}}
|}

Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by {{mvar|N<sub>n</sub>}} and {{mvar|n<sub>n</sub>}}, respectively, rather than {{mvar|Y<sub>n</sub>}} and {{mvar|y<sub>n</sub>}}.<ref>{{MathWorld|id=SphericalBesselFunctionoftheSecondKind|title=Spherical Bessel Function of the Second Kind}}</ref><ref name="MathWorld"/>

=== Bessel functions of the first kind: ''J<sub>α</sub>'' <span class="anchor" id="Bessel functions of the first kind"></span> ===
[[File:BesselJ.png|thumb|350px|right|Plot of Bessel function of the first kind, <math>J_\alpha(x)</math>, for integer orders <math>\alpha=0,1,2</math>.]]
[[File:Bessel half.png|thumb|350px|right|Plot of Bessel function of the first kind <math>J_\alpha(z)</math> with <math>\alpha=0.5</math> in the plane from <math>-4-4i</math> to <math>4+4i</math>.]]

Bessel functions of the first kind, denoted as {{math|''J<sub>α</sub>''(''x'')}}, are solutions of Bessel's differential equation. For integer or positive&nbsp;{{mvar|α}}, Bessel functions of the first kind are finite at the origin ({{math|1=''x'' = 0}}); while for negative non-integer&nbsp;{{mvar|α}}, Bessel functions of the first kind diverge as {{mvar|x}} approaches zero. It is possible to define the function by <math>x^\alpha</math> times a [[Maclaurin series]] (note that {{mvar|α}} need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying the [[Frobenius method]] to Bessel's equation:<ref name=p360>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_360.htm p. 360, 9.1.10].</ref>
<math display="block"> J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m + \alpha},</math>
where {{math|Γ(''z'')}} is the [[gamma function]], a shifted generalization of the [[factorial]] function to non-integer values. Some earlier authors define the Bessel function of the first kind differently, essentially without the division by <math>2</math> in <math>x/2</math>;<ref>{{cite book |last1=Whittaker |first1=Edmund Taylor |authorlink1=Edmund T. Whittaker |last2=Watson |first2=George Neville |authorlink2=George N. Watson |date= 1927 |page=356 |edition=4th |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |publisher= Cambridge University Press}} For example, Hansen (1843) and Schlömilch (1857).</ref> this definition is not used in this article. The Bessel function of the first kind is an [[entire function]] if {{mvar|α}} is an integer, otherwise it is a [[multivalued function]] with singularity at zero. The graphs of Bessel functions look roughly like oscillating [[Sine function|sine]] or [[cosine]] functions that decay proportionally to <math>x^{-{1}/{2}}</math> (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large {{mvar|x}}. (The series indicates that {{math|−''J''<sub>1</sub>(''x'')}} is the derivative of {{math|''J''<sub>0</sub>(''x'')}}, much like {{math|−sin ''x''}} is the derivative of {{math|cos ''x''}}; more generally, the derivative of {{math|''J<sub>n</sub>''(''x'')}} can be expressed in terms of {{math|''J''<sub>''n'' ± 1</sub>(''x'')}} by the identities [[#Properties|below]].)

For non-integer {{mvar|α}}, the functions {{math|''J<sub>α</sub>''(''x'')}} and {{math|''J''<sub>−''α''</sub>(''x'')}} are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order {{mvar|n}}, the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_358.htm p. 358, 9.1.5].</ref>
<math display="block">J_{-n}(x) = (-1)^n J_n(x).</math>

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.

==== Bessel's integrals ====
Another definition of the Bessel function, for integer values of {{mvar|n}}, is possible using an integral representation:<ref name=Temme>{{cite book |last=Temme |first=Nico M. |title=Special Functions: An introduction to the classical functions of mathematical physics |year=1996 |publisher=Wiley |location=New York |isbn=0471113131 |pages=228–231 |edition=2nd print}}</ref>
<math display="block">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),</math>
which is also called Hansen-Bessel formula.<ref>{{MathWorld|id=Hansen-BesselFormula|title=Hansen-Bessel Formula}}</ref>

This was the approach that Bessel used,<ref>Bessel, F. (1824).  The relevant integral is an unnumbered equation between equations 28 and 29.  Note that Bessel's <math>I^h_k</math> would today be written <math>J_h(k)</math>.</ref> 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 {{math|Re(''x'') > 0}}:<ref name=Temme /><ref>Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA176 p. 176]</ref><ref>{{cite web |url=http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html |title=Properties of Hankel and Bessel Functions |access-date=2010-10-18 |url-status=dead |archive-url=https://web.archive.org/web/20100923194031/http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html |archive-date=2010-09-23}}</ref><ref>{{cite web |url=https://www.nbi.dk/~polesen/borel/node15.html |title=Integral representations of the Bessel function |website=www.nbi.dk |access-date=25 March 2018 |archive-date=3 October 2022 |archive-url=https://web.archive.org/web/20221003054117/https://www.nbi.dk/~polesen/borel/node15.html |url-status=dead }}</ref><ref>Arfken & Weber, exercise 11.1.17.</ref>
<math display="block">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. </math>

==== Relation to hypergeometric series ====
The Bessel functions can be expressed in terms of the [[generalized hypergeometric series]] as<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_362.htm p. 362, 9.1.69].</ref>
<math display="block">J_\alpha(x) = \frac{\left(\frac{x}{2}\right)^\alpha}{\Gamma(\alpha+1)} \;_0F_1 \left(\alpha+1; -\frac{x^2}{4}\right).</math>

This expression is related to the development of Bessel functions in terms of the [[Bessel–Clifford function]].

==== Relation to Laguerre polynomials ====
In terms of the [[Laguerre polynomials]] {{mvar|L<sub>k</sub>}} and arbitrarily chosen parameter {{mvar|t}}, the Bessel function can be expressed as<ref>{{cite book |author-link=Gábor Szegő |last=Szegő |first=Gábor |title=Orthogonal Polynomials |edition=4th |location=Providence, RI |publisher=AMS |date=1975}}</ref>
<math display="block">\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}{4 t}\right)}{\binom{k+\alpha}{k}} \frac{t^k}{k!}.</math>

=== Bessel functions of the second kind: ''Y<sub>α</sub>'' <span class="anchor" id="Weber functions"></span><span class="anchor" id="Neumann functions"></span><span class="anchor" id="Bessel functions of the second kind"></span> ===
[[File:Besselyn.png|thumb|350px|Plot of Bessel function of the second kind, <math>Y_\alpha(x)</math>, for integer orders <math>\alpha = 0, 1, 2</math>]]
The Bessel functions of the second kind, denoted by {{math|''Y<sub>α</sub>''(''x'')}}, occasionally denoted instead by {{math|''N<sub>α</sub>''(''x'')}}, are solutions of the Bessel differential equation that have a singularity at the origin ({{math|1=''x'' = 0}}) and are [[multivalued function|multivalued]]. These are sometimes called '''Weber functions''', as they were introduced by {{harvs|txt|authorlink=Heinrich Martin Weber|first=H. M.|last=Weber|year=1873}}, and also '''Neumann functions''' after [[Carl Neumann]].<ref name="mhtlab.uwaterloo.ca">{{cite web |url=http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf |archive-date=2022-10-09 |url-status=live |title=Bessel Functions of the First and Second Kind |website=mhtlab.uwaterloo.ca |access-date=24 May 2022 |page=3}}</ref>

For non-integer {{mvar|α}}, {{math|''Y<sub>α</sub>''(''x'')}} is related to {{math|''J<sub>α</sub>''(''x'')}} by
<math display="block">Y_\alpha(x) = \frac{J_\alpha(x) \cos (\alpha \pi) - J_{-\alpha}(x)}{\sin (\alpha \pi)}.</math>

In the case of integer order {{mvar|n}}, the function is defined by taking the limit as a non-integer {{mvar|α}} tends to {{mvar|n}}:
<math display="block">Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x).</math>

If {{mvar|n}} is a nonnegative integer, we have the series<ref>[https://dlmf.nist.gov/10.8#E1 NIST Digital Library of Mathematical Functions], (10.8.1). Accessed on line Oct. 25, 2016.</ref>
<math display="block">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)!}</math>
where <math>\psi(z)</math> is the [[digamma function]], the [[logarithmic derivative]] of the [[gamma function]].<ref name="MathWorld">{{MathWorld|id=BesselFunctionoftheSecondKind|title=Bessel Function of the Second Kind}}</ref>

There is also a corresponding integral formula (for {{math|Re(''x'') > 0}}):<ref name="p. 178">Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA178 p. 178].</ref>
<math display="block">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.</math>

In the case where {{math|''n'' {{=}} 0}}: (with <math>\gamma</math> being [[Euler's constant]])<math display="block">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.</math>

[[File:Besselyhalf.png|thumb|300px|Plot of the Bessel function of the second kind <math>Y_\alpha(z)</math> with <math>\alpha = 0.5</math> in the complex plane from <math> -2 -2i</math> to <math>2 + 2i</math>.]]
{{math|''Y<sub>α</sub>''(''x'')}} is necessary as the second linearly independent solution of the Bessel's equation when {{mvar|α}} is an integer. But {{math|''Y<sub>α</sub>''(''x'')}} has more meaning than that. It can be considered as a "natural" partner of {{math|''J<sub>α</sub>''(''x'')}}. See also the subsection on Hankel functions below.

When {{mvar|α}} is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:
<math display="block">Y_{-n}(x) = (-1)^n Y_n(x).</math>

Both {{math|''J<sub>α</sub>''(''x'')}} and {{math|''Y<sub>α</sub>''(''x'')}} are [[holomorphic function]]s of {{mvar|x}} on the [[complex plane]] cut along the negative real axis. When {{mvar|α}} is an integer, the Bessel functions {{mvar|J}} are [[entire function]]s of {{mvar|x}}. If {{mvar|x}} is held fixed at a non-zero value, then the Bessel functions are entire functions of {{mvar|α}}.

The Bessel functions of the second kind when {{mvar|α}} is an integer is an example of the second kind of solution in [[Fuchs's theorem]].

=== Hankel functions: ''H''{{su|b=''α''|p=(1)}}, ''H''{{su|b=''α''|p=(2)}} <span class="anchor" id="Hankel functions"></span> ===
[[File:Plot of the Hankel function of the first kind H n^(1)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the Hankel function of the first kind {{math|''H''{{su|b=''n''|p=(1)}}(''x'')}} with {{math|1=''n'' = −0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
[[File:Plot of the Hankel function of the second kind H n^(2)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the Hankel function of the second kind {{math|''H''{{su|b=''n''|p=(2)}}(''x'')}} with {{math|1=''n'' = −0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]

Another important formulation of the two linearly independent solutions to Bessel's equation are the '''Hankel functions of the first and second kind''', {{math|''H''{{su|b=''α''|p=(1)}}(''x'')}} and {{math|''H''{{su|b=''α''|p=(2)}}(''x'')}}, defined as<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_358.htm p. 358, 9.1.3, 9.1.4].</ref>
<math display="block">\begin{align}
H_\alpha^{(1)}(x) &= J_\alpha(x) + iY_\alpha(x), \\[5pt]
H_\alpha^{(2)}(x) &= J_\alpha(x) - iY_\alpha(x),
\end{align}</math>
where {{mvar|i}} is the [[imaginary unit]]. 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 [[Hermann Hankel]].

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 {{math|''e''<sup>''i'' ''f''(x)</sup>}}. For real <math>x>0</math> where <math>J_\alpha(x)</math>, <math>Y_\alpha(x)</math> 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 [[Euler's formula]], substituting {{math|''H''{{su|b=''α''|p=(1)}}(''x'')}}, {{math|''H''{{su|b=''α''|p=(2)}}(''x'')}} for <math>e^{\pm i x}</math> and <math>J_\alpha(x)</math>, <math>Y_\alpha(x)</math> for <math>\cos(x)</math>, <math>\sin(x)</math>, as explicitly shown in the [[#Asymptotic forms|asymptotic expansion]].

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 [[sign convention]] for the [[frequency]]).

Using the previous relationships, they can be expressed as
<math display="block">\begin{align}
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{align}</math>

If {{mvar|α}} is an integer, the limit has to be calculated. The following relationships are valid, whether {{mvar|α}} is an integer or not:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_358.htm p. 358, 9.1.6].</ref>
<math display="block">\begin{align}
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{align}</math>

In particular, if {{math|1=''α'' = ''m'' + {{sfrac|1|2}}}} with {{mvar|m}} a nonnegative integer, the above relations imply directly that
<math display="block">\begin{align}
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{align}</math>

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

The Hankel functions admit the following integral representations for {{math|Re(''x'') > 0}}:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_360.htm p. 360, 9.1.25].</ref>
<math display="block">\begin{align}
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{align}</math>
where the integration limits indicate integration along a [[methods of contour integration|contour]] that can be chosen as follows: from {{math|−∞}} to 0 along the negative real axis, from 0 to {{math|±{{pi}}''i''}} along the imaginary axis, and from {{math|±{{pi}}''i''}} to {{math|+∞ ± {{pi}}''i''}} along a contour parallel to the real axis.<ref name="p. 178"/>

=== Modified Bessel functions: ''I<sub>α</sub>'', ''K<sub>α</sub>'' <span class="anchor" id="Modified Bessel functions"></span><span class="anchor" id="Modified Bessel functions : Iα, Kα"></span> ===
The Bessel functions are valid even for [[complex number|complex]] arguments {{mvar|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<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_375.htm p. 375, 9.6.2, 9.6.10, 9.6.11].</ref>
<math display="block">\begin{align}
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{align}</math>
when {{mvar|α}} is not an integer. When {{mvar|α}} is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments {{mvar|x}}. The series expansion for {{math|''I<sub>α</sub>''(''x'')}} is thus similar to that for {{math|''J<sub>α</sub>''(''x'')}}, but without the alternating {{math|(−1)<sup>''m''</sup>}} factor.

<math>K_{\alpha}</math> can be expressed in terms of Hankel functions:
<math display="block">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}</math>

Using these two formulae the result to <math>J_{\alpha}^2(z)</math>+<math>Y_{\alpha}^2(z)</math>, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following
<math display="block">
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,
</math>

given that the condition {{math|Re(''x'') > 0}} is met. It can also be shown that
<math display="block">
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,
</math>
only when {{math|{{abs|Re(''α'')}} < {{sfrac|1|2}}}} and {{math|Re(''x'') &ge; 0}} but not when {{math|1=''x'' = 0}}.<ref>{{cite journal |last1=Dixon |last2=Ferrar |first2=W.L. |date=1930 |title=A direct proof of Nicholson's integral |journal=The Quarterly Journal of Mathematics |location=Oxford |pages=236–238 |doi=10.1093/qmath/os-1.1.236}}</ref>

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if {{math|−''π'' < arg ''z'' ≤ {{sfrac|''π''|2}}}}):<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_375.htm p. 375, 9.6.3, 9.6.5].</ref>
<math display="block">\begin{align}
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{align}</math>

{{math|''I<sub>α</sub>''(''x'')}} and {{math|''K<sub>α</sub>''(''x'')}} are the two linearly independent solutions to the '''modified Bessel's equation''':<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_374.htm p. 374, 9.6.1].</ref>
<math display="block">x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - \left(x^2 + \alpha^2 \right)y = 0.</math>

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, {{mvar|I<sub>α</sub>}} and {{mvar|K<sub>α</sub>}} are [[exponential growth|exponentially growing]] and [[exponential decay|decaying]] functions respectively. Like the ordinary Bessel function {{mvar|J<sub>α</sub>}}, the function {{mvar|I<sub>α</sub>}} goes to zero at {{math|1=''x'' = 0}} for {{math|''α'' > 0}} and is finite at {{math|1=''x'' = 0}} for {{math|1=''α'' = 0}}. Analogously, {{mvar|K<sub>α</sub>}} diverges at {{math|1=''x'' = 0}} with the singularity being of logarithmic type for {{mvar|K<sub>0</sub>}}, and {{math|1={{sfrac|1|2}}Γ({{abs|''α''}})(2/''x'')<sup>{{abs|''α''}}</sup>}} otherwise.<ref>{{cite book |title=Quantum Electrodynamics |last1=Greiner |first1=Walter |last2=Reinhardt |first2=Joachim |date=2009 |publisher=Springer |page=72 |isbn=978-3-540-87561-1}}</ref>

{|
| [[File:Besseli.png|none|thumb|350px|Modified Bessel functions of the first kind, <math>I_\alpha(x)</math>, for <math>\alpha = 0, 1, 2, 3</math>.]]
| [[File:Besselk.png|none|thumb|350px|Modified Bessel functions of the second kind, <math>K_\alpha(x)</math>, for <math>\alpha = 0, 1, 2, 3</math>.]]
|}
<!-- [[File:ModifiedBessel.png|Plot of some modified Bessel functions]]<br />Plot of six modified Bessel functions. In solid line {{math|''K''<sub>0</sub>}}, {{math|''K''<sub>1</sub>}}, and {{math|''K''<sub>2</sub>}}. In dashed line: {{math|''I''<sub>0</sub>}}, {{math|''I''<sub>1</sub>}}, and {{math|''I''<sub>2</sub>}}. -->

Two integral formulas for the modified Bessel functions are (for {{math|Re(''x'') > 0}}):<ref>Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA181 p. 181].</ref>
<math display="block">\begin{align}
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{align}</math>

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for {{math|Re(ω) > 0}}):
<math display="block">2\,K_0(\omega) = \int_{-\infty}^\infty \frac{e^{i\omega t}}{\sqrt{t^2+1}} \,dt.</math>

It can be proven by showing equality to the above integral definition for {{math|''K''<sub>0</sub>}}. 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 <ref>{{cite web |url=http://dlmf.nist.gov/10.32.E11
|title=Modified Bessel Functions §10.32 Integral Representations |author=<!--Not stated--> |date=<!--Not stated--> |website=NIST Digital Library of Mathematical Functions |publisher=NIST |access-date=2024-11-20}}</ref>
<math display="block"> 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}}}.</math>

Modified Bessel functions {{math|''K''<sub>1/3</sub>}} and {{math|''K''<sub>2/3</sub>}} can be represented in terms of rapidly convergent integrals<ref>{{cite journal |first=M. Kh. |last=Khokonov |title=Cascade Processes of Energy Loss by Emission of Hard Photons |journal=Journal of Experimental and Theoretical Physics |volume=99 |issue=4 |pages=690–707 |date=2004 |doi=10.1134/1.1826160 |bibcode=2004JETP...99..690K |s2cid=122599440}}. Derived from formulas sourced to [[Gradshteyn and Ryzhik|I. S. Gradshteyn and I. M. Ryzhik]], ''[[Table of Integrals, Series, and Products]]'' (Fizmatgiz, Moscow, 1963; Academic Press, New York, 1980).</ref>
<math display="block"> \begin{align}
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{align}</math>

The modified Bessel function <math>K_{\frac{1}{2}}(\xi)=(2 \xi / \pi)^{-1/2}\exp(-\xi)</math> is useful to represent the Laplace distribution 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):
* '''Basset function''' after [[Alfred Barnard Basset]]
* '''Modified Bessel function of the third kind'''
* '''Modified Hankel function'''<ref>Referred to as such in: {{cite journal |last=Teichroew |first=D. |title=The Mixture of Normal Distributions with Different Variances |journal=The Annals of Mathematical Statistics |volume=28 |issue=2 |date=1957 |pages=510–512 |doi=10.1214/aoms/1177706981 |url=https://dml.cz/bitstream/handle/10338.dmlcz/103973/AplMat_27-1982-4_7.pdf |doi-access=free}}</ref>
* '''Macdonald function''' after [[Hector Munro Macdonald]]

=== Spherical Bessel functions: ''j<sub>n</sub>'', ''y<sub>n</sub>'' <span class="anchor" id="Spherical Bessel functions"></span> ===
[[File:Plot of the spherical Bessel function of the first kind j n(z) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the spherical Bessel function of the first kind {{math|''j<sub>n</sub>''(''z'')}} with {{math|1=''n'' = 0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]
[[File:Plot of the spherical Bessel function of the second kind y n(z) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the spherical Bessel function of the second kind {{math|''y<sub>n</sub>''(''z'')}} with {{math|1=''n'' = 0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]

[[File:Sphericalbesselj.png|thumb|350px|right|Spherical Bessel functions of the first kind <math> j_\alpha(x)</math>, for <math>\alpha = 0,1,2</math>.]]
[[File:Sphericalbessely.png|thumb|350px|right|Spherical Bessel functions of the second kind <math> y_\alpha(x)</math>, for <math>\alpha = 0,1,2</math>.]]

When solving the [[Helmholtz equation]] in spherical coordinates by separation of variables, the radial equation has the form
<math display="block">x^2 \frac{d^2 y}{dx^2} + 2x \frac{d y}{dx} +\left(x^2 - n(n + 1)\right) y = 0.</math>

The two linearly independent solutions to this equation are called the '''spherical Bessel functions''' {{mvar|j<sub>n</sub>}} and {{mvar|y<sub>n</sub>}}, and are related to the ordinary Bessel functions {{mvar|J<sub>n</sub>}} and {{mvar|Y<sub>n</sub>}} by<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_437.htm p. 437, 10.1.1].</ref>
<math display="block">\begin{align}
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{align}</math>

{{mvar|y<sub>n</sub>}} is also denoted {{mvar|n<sub>n</sub>}} or {{mvar|[[Eta (letter)|η]]<sub>n</sub>}}; some authors call these functions the '''spherical Neumann functions'''.

From the relations to the ordinary Bessel functions it is directly seen that:
<math display="block">\begin{align}
j_n(x) &= (-1)^{n} y_{-n-1} (x)
\\
y_n(x) &= (-1)^{n+1} j_{-n-1}(x)
\end{align}</math>

The spherical Bessel functions can also be written as ('''{{va|Rayleigh's formulas}}''')<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.25, 10.1.26].</ref>
<math display="block">\begin{align}
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{align}</math>

The zeroth spherical Bessel function {{math|''j''<sub>0</sub>(''x'')}} is also known as the (unnormalized) [[sinc function]]. The first few spherical Bessel functions are:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_438.htm p. 438, 10.1.11].</ref>
<math display="block">\begin{align}
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{align}</math>
and<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_438.htm p. 438, 10.1.12].</ref>
<math display="block">\begin{align}
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{align}</math>

The first few non-zero roots of the first few spherical Bessel functions are:
{| class="wikitable sortable"
|+ Non-zero Roots of the Spherical Bessel Function (first kind)
! Order !! Root 1 !! Root 2 !! Root 3 !! Root 4 !! Root 5
|-
| <math>j_{0}</math> || 3.141593 || 6.283185 || 9.424778 || 12.566371 || 15.707963
|-
| <math>j_{1}</math> || 4.493409 || 7.725252 || 10.904122 || 14.066194 || 17.220755
|-
| <math>j_{2}</math> || 5.763459 || 9.095011 || 12.322941 || 15.514603 || 18.689036
|-
| <math>j_{3}</math> || 6.987932 || 10.417119 || 13.698023 || 16.923621 || 20.121806
|-
| <math>j_{4}</math> || 8.182561 || 11.704907 || 15.039665 || 18.301256 || 21.525418
|}

{| class="wikitable sortable"
|+ Non-zero Roots of the Spherical Bessel Function (second kind)
! Order !! Root 1 !! Root 2 !! Root 3 !! Root 4 !! Root 5
|-
| <math>y_{0}</math> || 1.570796 || 4.712389 || 7.853982 || 10.995574 || 14.137167
|-
| <math>y_{1}</math> || 2.798386 || 6.121250 || 9.317866 || 12.486454 || 15.644128
|-
| <math>y_{2}</math> || 3.959528 || 7.451610 || 10.715647 || 13.921686 || 17.103359
|-
| <math>y_{3}</math> || 5.088498 || 8.733710 || 12.067544 || 15.315390 || 18.525210
|-
| <math>y_{4}</math> || 6.197831 || 9.982466 || 13.385287 || 16.676625 || 19.916796
|}

==== Generating function ====
The spherical Bessel functions have the generating functions<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.39].</ref>
<math display="block">\begin{align}
\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{align}</math>

==== Finite series expansions ====
In contrast to the whole integer Bessel functions {{math|''J''<sub>n</sub>(''x''), ''Y''<sub>n</sub>(''x'')}}, the spherical Bessel functions {{math|''j''<sub>n</sub>(''x''), ''y''<sub>n</sub>(''x'')}} have a finite series expression:<ref>L.V. Babushkina, M.K. Kerimov, A.I. Nikitin, Algorithms for computing Bessel functions of half-integer order with complex arguments, [https://www.sciencedirect.com/science/article/pii/0041555388900183 p. 110, p. 111].</ref>
<math display="block">\begin{alignat}{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{alignat}</math>

==== Differential relations ====
In the following, {{mvar|f<sub>n</sub>}} is any of {{mvar|j<sub>n</sub>}}, {{mvar|y<sub>n</sub>}}, {{math|''h''{{su|b=''n''|p=(1)}}}}, {{math|''h''{{su|b=''n''|p=(2)}}}} for {{math|1=''n'' = 0, ±1, ±2, ...}}<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.23, 10.1.24].</ref>
<math display="block">\begin{align}
\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{align}</math>

=== Spherical Hankel functions: ''h''{{su|b=''n''|p=(1)}}, ''h''{{su|b=''n''|p=(2)}} <span class="anchor" id="Spherical Hankel functions"></span> ===
[[File:Plot of the spherical Hankel function of the first kind h n^(1)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the spherical Hankel function of the first kind {{math|''h''{{su|b=''n''|p=(1)}}(''x'')}} with {{math|1=''n'' = −0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
[[File:Plot of the spherical Hankel function of the second kind h n^(2)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the spherical Hankel function of the second kind {{math|''h''{{su|b=''n''|p=(2)}}(''x'')}} with {{math|1=''n'' = −0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]

There are also spherical analogues of the [[#Hankel functions|Hankel functions]]:
<math display="block">\begin{align}
h_n^{(1)}(x) &= j_n(x) + i y_n(x), \\
h_n^{(2)}(x) &= j_n(x) - i y_n(x).
\end{align}</math>

There are simple closed-form expressions for the Bessel functions of [[half-integer]] order in terms of the standard [[trigonometric function]]s, and therefore for the spherical Bessel functions. In particular, for non-negative integers {{mvar|n}}:
<math display="block">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)!},</math>
and {{math|''h''{{su|b=''n''|p=(2)}}}} is the complex-conjugate of this (for real {{mvar|x}}). It follows, for example, that {{math|1=''j''<sub>0</sub>(''x'') = {{sfrac|sin ''x''|''x''}}}} and {{math|1=''y''<sub>0</sub>(''x'') = −{{sfrac|cos ''x''|''x''}}}}, and so on.

The spherical Hankel functions appear in problems involving [[spherical wave]] propagation, for example in the [[electromagnetic wave equation#Multipole expansion|multipole expansion of the electromagnetic field]].

=== Riccati–Bessel functions: ''S<sub>n</sub>'', ''C<sub>n</sub>'', ''ξ<sub>n</sub>'', ''ζ<sub>n</sub>'' <span class="anchor" id="Riccati–Bessel functions"></span> ===
[[Jacopo Riccati|Riccati]]–Bessel functions only slightly differ from spherical Bessel functions:
<math display="block">\begin{align}
S_n(x)     &= x j_n(x) = \sqrt{\frac{\pi x}{2}} J_{n+\frac{1}{2}}(x) \\
C_n(x)     &= -x y_n(x) = -\sqrt{\frac{\pi x}{2}} Y_{n+\frac{1}{2}}(x) \\
\xi_n(x)   &= x h_n^{(1)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(1)}(x) = S_n(x) - iC_n(x) \\
\zeta_n(x) &= x h_n^{(2)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(2)}(x) = S_n(x) + iC_n(x)
\end{align}</math>
[[File:Riccati Bessel Function S 3D Complex Color Plot with Mathematica 13.2.svg|alt=Riccati–Bessel functions  Sn complex plot from -2-2i to 2+2i|thumb|Riccati–Bessel functions Sn complex plot from −2&nbsp;−&nbsp;2''i'' to 2&nbsp;+&nbsp;2''i'']]
They satisfy the differential equation
<math display="block">x^2 \frac{d^2 y}{dx^2} + \left (x^2 - n(n + 1)\right) y = 0.</math>

For example, this kind of differential equation appears in [[quantum mechanics]] while solving the radial component of the [[Schrödinger equation]] with hypothetical cylindrical infinite potential barrier.<ref>Griffiths. Introduction to Quantum Mechanics, 2nd edition, p.&nbsp;154.</ref> This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known as [[Mie scattering]] after the first published solution by Mie (1908). See e.g., Du (2004)<ref>{{cite journal |first=Hong |last=Du |title=Mie-scattering calculation |journal=Applied Optics |volume=43 |issue=9 |pages=1951–1956 |date=2004 |doi=10.1364/ao.43.001951 |pmid=15065726 |bibcode=2004ApOpt..43.1951D}}</ref> for recent developments and references.

Following [[Peter Debye|Debye]] (1909), the notation {{mvar|ψ<sub>n</sub>}}, {{mvar|χ<sub>n</sub>}} is sometimes used instead of {{mvar|S<sub>n</sub>}}, {{mvar|C<sub>n</sub>}}.

== Asymptotic forms ==
The Bessel functions have the following [[asymptotic analysis|asymptotic]] forms. For small arguments <math>0<z\ll\sqrt{\alpha+1}</math>, one obtains, when <math>\alpha</math> is not a negative integer:<ref name=p360/>
<math display="block">J_\alpha(z) \sim \frac{1}{\Gamma(\alpha+1)} \left( \frac{z}{2} \right)^\alpha.</math>

When {{mvar|α}} is a negative integer, we have
<math display="block">J_\alpha(z) \sim \frac{(-1)^{\alpha}}{(-\alpha)!} \left( \frac{2}{z} \right)^\alpha.</math>

For the Bessel function of the second kind we have three cases:
<math display="block">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}</math>
where {{mvar|γ}} is the [[Euler–Mascheroni constant]] (0.5772...).

For large real arguments {{math|''z'' ≫ {{abs|''α''<sup>2</sup> − {{sfrac|1|4}}}}}}, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless {{mvar|α}} is [[half-integer]]) because they have [[zero of a function|zeros]] all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of {{math|arg ''z''}} one can write an equation containing a term of order {{math|{{abs|''z''}}<sup>−1</sup>}}:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_364.htm p. 364, 9.2.1].</ref>
<math display="block">\begin{align}
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{align}</math>

(For {{math|1=''α'' = {{sfrac|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:
<math display="block">\begin{align}
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{align}</math>

These can be extended to other values of {{math|arg ''z''}} using equations relating {{math|''H''{{su|b=''α''|p=(1)}}(''ze''<sup>''im''π</sup>)}} and {{math|''H''{{su|b=''α''|p=(2)}}(''ze''<sup>''im''π</sup>)}} to {{math|''H''{{su|b=''α''|p=(1)}}(''z'')}} and {{math|''H''{{su|b=''α''|p=(2)}}(''z'')}}.<ref>[[NIST]] [[Digital Library of Mathematical Functions]], Section [https://dlmf.nist.gov/10.11#E1 10.11].</ref>

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, {{math|''J<sub>α</sub>''(''z'')}} is not asymptotic to the average of these two asymptotic forms when {{mvar|z}} is negative (because one or the other will not be correct there, depending on the {{math|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) {{mvar|z}} so long as {{math|{{abs|''z''}}}} goes to infinity at a constant phase angle {{math|arg ''z''}} (using the square root having positive real part):
<math display="block">\begin{align}
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{align}</math>

For the modified Bessel functions, [[Hermann Hankel|Hankel]] developed [[asymptotic expansion]]s as well:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_377.htm p. 377, 9.7.1].</ref><ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_378.htm p. 378, 9.7.2].</ref>
<math display="block">\begin{align}
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{align}</math>

There is also the asymptotic form (for large real <math>z</math>)<ref>[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]</ref>
<math display="block">\begin{align}
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{align}</math>

When {{math|1=''α'' = {{sfrac|1|2}}}}, all the terms except the first vanish, and we have
<math display="block">\begin{align}
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{align}</math>

For small arguments <math>0<|z|\ll\sqrt{\alpha + 1}</math>, we have
<math display="block">\begin{align}
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{align}</math>

== Properties ==
<!-- This section is linked from [[Bessel function]] -->
For integer order {{math|1=''α'' = ''n''}}, {{mvar|J<sub>n</sub>}} is often defined via a [[Laurent series]] for a generating function:
<math display="block">e^{\frac{x}{2}\left(t-\frac{1}{t}\right)} = \sum_{n=-\infty}^\infty J_n(x) t^n</math>
an approach used by [[P. A. Hansen]] in 1843. (This can be generalized to non-integer order by [[methods of contour integration|contour integration]] or other methods.)

Infinite series of Bessel functions in the form <math display="inline"> \sum_{\nu=-\infty}^\infty J_{N\nu + p}(x)</math> where <math display>\nu, p \in \mathbb{Z}, \ N \in \mathbb{Z}^+</math> arise in many physical systems and are defined in closed form by the [[Sung series]].<ref name="SungSeries">{{cite arXiv |last1=Sung |first1=S. |last2=Hovden |first2=R. |title=On Infinite Series of Bessel functions of the First Kind |year=2022 |class=math-ph |eprint=2211.01148}}</ref> For example, when N = 3: <math display="inline"> \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] </math>. More generally, the Sung series and the alternating Sung series are written as:
<math display = "block"> \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}
</math>
<math display = "block"> \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}
</math>

A series expansion using Bessel functions ([[Kapteyn series]]) is
<math display="block">\frac {1}{1-z} = 1 + 2 \sum _{n=1}^{\infty } J_{n}(nz).</math>

Another important relation for integer orders is the ''[[Jacobi–Anger expansion]]'':
<math display="block">e^{iz \cos \phi} = \sum_{n=-\infty}^\infty i^n J_n(z) e^{in\phi}</math>
and
<math display="block">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)</math>
which is used to expand a [[plane wave]] as a [[plane wave expansion|sum of cylindrical waves]], or to find the [[Fourier series]] of a tone-modulated [[frequency modulation|FM]] signal.

More generally, a series
<math display="block">f(z)=a_0^\nu J_\nu (z)+ 2 \cdot \sum_{k=1}^\infty a_k^\nu J_{\nu+k}(z)</math>
is called Neumann expansion of {{mvar|f}}. The coefficients for {{math|1=''ν'' = 0}} have the explicit form
<math display="block">a_k^0=\frac{1}{2 \pi i} \int_{|z|=c} f(z) O_k(z) \,dz</math>
where {{mvar|O<sub>k</sub>}} is [[Neumann polynomial|Neumann's polynomial]].<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_363.htm p. 363, 9.1.82] ff.</ref>

Selected functions admit the special representation
<math display="block">f(z)=\sum_{k=0}^\infty a_k^\nu J_{\nu+2k}(z)</math>
with
<math display="block">a_k^\nu=2(\nu+2k) \int_0^\infty f(z) \frac{J_{\nu+2k}(z)}z \,dz</math>
due to the orthogonality relation
<math display="block">\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}</math>

More generally, if {{mvar|f}} has a branch-point near the origin of such a nature that
<math display="block">f(z)= \sum_{k=0} a_k J_{\nu+k}(z)</math>
then
<math display="block">\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}}</math>
or
<math display="block">\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)</math>
where <math>\mathcal{L}\{f \}</math> is the [[Laplace transform]] of {{mvar|f}}.<ref>{{cite book |url=https://books.google.com/books?id=Mlk3FrNoEVoC&q=bessel+neumann+series&pg=PA536 |title=A Treatise on the Theory of Bessel Functions |first=G. N. |last=Watson |date=25 August 1995 |publisher=Cambridge University Press |access-date=25 March 2018 |via=Google Books |isbn=9780521483919}}</ref>

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:
<math display="block">\begin{align}
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{align}</math>
where {{math|ν > −{{sfrac|1|2}}}} and {{math|''z'' ∈ '''C'''}}.<ref name="Zwillinger_2014">{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=en |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21 --> |title-link=Gradshteyn and Ryzhik |chapter=8.411.10.}}</ref>
This formula is useful especially when working with [[Fourier transforms]].

Because Bessel's equation becomes [[Hermitian]] (self-adjoint) if it is divided by {{mvar|x}}, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:
<math display="block">\int_0^1 x J_\alpha\left(x u_{\alpha,m}\right) J_\alpha\left(x u_{\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</math>
where {{math|''α'' > −1}}, {{math|''δ''<sub>''m'',''n''</sub>}} is the [[Kronecker delta]], and {{math|''u''<sub>''α'',''m''</sub>}} is the {{mvar|m}}th [[root of a function|zero]] of {{math|''J<sub>α</sub>''(''x'')}}. This orthogonality relation can then be used to extract the coefficients in the [[Fourier–Bessel series]], where a function is expanded in the basis of the functions {{math|''J<sub>α</sub>''(''x'' ''u''<sub>''α'',''m''</sub>)}} for fixed {{mvar|α}} and varying {{mvar|m}}.

An analogous relationship for the spherical Bessel functions follows immediately:
<math display="block">\int_0^1 x^2 j_\alpha\left(x u_{\alpha,m}\right) j_\alpha\left(x u_{\alpha,n}\right) \,dx = \frac{\delta_{m,n}}{2} \left[j_{\alpha+1}\left(u_{\alpha,m}\right)\right]^2</math>

If one defines a [[boxcar function]] of {{mvar|x}} that depends on a small parameter {{mvar|ε}} as:
<math display="block">f_\varepsilon(x)=\frac 1\varepsilon \operatorname{rect}\left(\frac{x-1}\varepsilon\right)</math>
(where {{math|rect}} is the [[rectangle function]]) then the [[Hankel transform]] of it (of any given order {{math|''α'' > −{{sfrac|1|2}}}}), {{math|''g<sub>ε</sub>''(''k'')}}, approaches {{math|''J<sub>α</sub>''(''k'')}} as {{mvar|ε}} approaches zero, for any given {{mvar|k}}. Conversely, the Hankel transform (of the same order) of {{math|''g<sub>ε</sub>''(''k'')}} is {{math|''f<sub>ε</sub>''(''x'')}}:
<math display="block">\int_0^\infty k J_\alpha(kx) g_\varepsilon(k) \,dk = f_\varepsilon(x)</math>
which is zero everywhere except near 1. As {{mvar|ε}} approaches zero, the right-hand side approaches {{math|''δ''(''x'' − 1)}}, where {{mvar|δ}} is the [[Dirac delta function]]. This admits the limit (in the [[distribution (mathematics)|distributional]] sense):
<math display="block">\int_0^\infty k J_\alpha(kx) J_\alpha(k) \,dk = \delta(x-1)</math>

A change of variables then yields the ''closure equation'':<ref>Arfken & Weber, section 11.2</ref>
<math display="block">\int_0^\infty x J_\alpha(ux) J_\alpha(vx) \,dx = \frac{1}{u} \delta(u - v)</math>
for {{math|''α'' > −{{sfrac|1|2}}}}. The Hankel transform can express a fairly arbitrary function{{Clarify|reason=This can probably be precisely qualified e.g. square integrable etc.|date=June 2018}} as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is:
<math display="block">\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) \,dx = \frac{\pi}{2uv} \delta(u - v)</math>
for {{math|''α'' > −1}}.

Another important property of Bessel's equations, which follows from [[Abel's identity]], involves the [[Wronskian]] of the solutions:
<math display="block">A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x}</math>
where {{mvar|A<sub>α</sub>}} and {{mvar|B<sub>α</sub>}} are any two solutions of Bessel's equation, and {{mvar|C<sub>α</sub>}} is a constant independent of {{mvar|x}} (which depends on α and on the particular Bessel functions considered). In particular,
<math display="block">J_\alpha(x) \frac{dY_\alpha}{dx} - \frac{dJ_\alpha}{dx} Y_\alpha(x) = \frac{2}{\pi x}</math>
and
<math display="block">I_\alpha(x) \frac{dK_\alpha}{dx} - \frac{dI_\alpha}{dx} K_\alpha(x) = -\frac{1}{x},</math>
for {{math|''α'' > −1}}.

For {{math|''α'' > −1}}, the even entire function of genus 1, {{math|''x''<sup>−''α''</sup>''J<sub>α</sub>''(''x'')}}, has only real zeros. Let
<math display="block">0<j_{\alpha,1}<j_{\alpha,2}<\cdots<j_{\alpha,n}<\cdots</math>
be all its positive zeros, then
<math display="block">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)</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.)

=== Recurrence relations ===
The functions {{mvar|J<sub>α</sub>}}, {{mvar|Y<sub>α</sub>}}, {{math|''H''{{su|b=''α''|p=(1)}}}}, and {{math|''H''{{su|b=''α''|p=(2)}}}} all satisfy the [[recurrence relation]]s<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_361.htm p. 361, 9.1.27].</ref>
<math display="block">\frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x)</math>
and
<math display="block"> 2\frac{dZ_\alpha (x)}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x),</math>
where {{mvar|Z}} denotes {{mvar|J}}, {{mvar|Y}}, {{math|''H''<sup>(1)</sup>}}, or {{math|''H''<sup>(2)</sup>}}. 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<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_361.htm p. 361, 9.1.30].</ref>
<math display="block">\begin{align}
\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{align}</math>

''Modified'' Bessel functions follow similar relations:
<math display="block">e^{\left(\frac{x}{2}\right)\left(t+\frac{1}{t}\right)} = \sum_{n=-\infty}^\infty I_n(x) t^n</math>
and
<math display="block">e^{z \cos \theta} = I_0(z) + 2\sum_{n=1}^\infty I_n(z) \cos n\theta</math>
and
<math display="block"> \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).</math>

The recurrence relation reads
<math display="block">\begin{align}
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{align}</math>
where {{mvar|C<sub>α</sub>}} denotes {{mvar|I<sub>α</sub>}} or {{math|''e''<sup>''αi''π</sup>''K<sub>α</sub>''}}. These recurrence relations are useful for discrete diffusion problems.

=== Transcendence ===
In 1929, [[Carl Ludwig Siegel]] proved that {{math|''J''<sub>''ν''</sub>(''x'')}}, {{math|''J''{{'}}<sub>''ν''</sub>(''x'')}}, and the [[logarithmic derivative]] {{math|{{sfrac|''J''{{'}}<sub>''ν''</sub>(''x'')|''J''<sub>''ν''</sub>(''x'')}}}} are [[transcendental number]]s when ''ν'' is rational and ''x'' is algebraic and nonzero.<ref>{{cite book |last1=Siegel |first1=Carl L. |title=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 |date=2014 |publisher=Scuola Normale Superiore |isbn=978-88-7642-520-2 |pages=81–138 |chapter-url=https://link.springer.com/chapter/10.1007/978-88-7642-520-2_2 |language=de |chapter=Über einige Anwendungen diophantischer Approximationen |doi=10.1007/978-88-7642-520-2_2}}</ref> The same proof also implies that <math> \Gamma(v+1)(2/x)^v J_{v}(x) </math> is transcendental under the same assumptions.<ref name="euclid">{{cite journal |last1=James |first1=R. D. |title=Review: Carl Ludwig Siegel, Transcendental numbers |journal=Bulletin of the American Mathematical Society |date=November 1950 |volume=56 |issue=6 |pages=523–526 |doi=10.1090/S0002-9904-1950-09435-X |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-56/issue-6/Review-Carl-Ludwig-Siegel-Transcendental-numbers/bams/1183515049.full|doi-access=free }}</ref>

=== Sums with Bessel functions ===
The product of two Bessel functions admits the following sum:
<math display="block">\sum_{\nu=-\infty}^\infty J_\nu(x) J_{n - \nu}(y) = J_{n}(x + y),</math>
<math display="block">\sum_{\nu=-\infty}^\infty J_\nu(x) J_{\nu + n}(y) = J_{n}(y - x).</math>
From these equalities it follows that
<math display="block">\sum_{\nu=-\infty}^\infty J_\nu(x) J_{\nu + n}(x) = \delta_{n, 0}</math>
and as a consequence
<math display="block">\sum_{\nu=-\infty}^\infty J_{\nu}^2(x) = 1. </math>

These sums can be extended to include a term multiplier that is a polynomial function of the index. For example,
<math display="block">\sum_{\nu=-\infty}^\infty \nu J_\nu(x) J_{\nu + n}(x) = \frac{x}{2} \left( \delta_{n, 1} + \delta_{n, -1} \right),</math>
<math display="block">\sum_{\nu=-\infty}^\infty \nu J_{\nu}^2(x) = 0, </math>
<math display="block">\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),</math>
<math display="block">\sum_{\nu=-\infty}^\infty \nu^2 J_{\nu}^2(x) = \frac{x^2}{2}. </math>

== Multiplication theorem ==
The Bessel functions obey a [[multiplication theorem]]
<math display="block">\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),</math>
where {{mvar|λ}} and {{mvar|ν}} may be taken as arbitrary complex numbers.<ref name=Abramowitz_9_1_74>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_363.htm p. 363, 9.1.74].</ref><ref>{{cite journal |last=Truesdell |first=C. |title=On the Addition and Multiplication Theorems for the Special Functions |journal=Proceedings of the National Academy of Sciences |volume=1950 |issue=12 |pages=752–757 |date=1950 |doi=10.1073/pnas.36.12.752 |pmid=16578355 |pmc=1063284 |bibcode=1950PNAS...36..752T |doi-access=free}}</ref> For {{math|{{abs|''λ''<sup>2</sup> − 1}} < 1}},<ref name=Abramowitz_9_1_74 /> the above expression also holds if {{mvar|J}} is replaced by {{mvar|Y}}. The analogous identities for modified Bessel functions and {{math|{{abs|''λ''<sup>2</sup> − 1}} < 1}} are
<math display="block">\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)</math>
and
<math display="block">\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).</math>

== Zeros of the Bessel function ==

=== Bourget's hypothesis ===
Bessel himself originally proved that for nonnegative integers {{mvar|n}}, the equation {{math|1=''J''<sub>''n''</sub>(''x'') = 0}} has an infinite number of solutions in {{mvar|x}}.<ref>Bessel, F. (1824), article 14.</ref> When the functions {{math|''J''<sub>''n''</sub>(''x'')}} are plotted on the same graph, though, none of the zeros seem to coincide for different values of {{mvar|n}} except for the zero at {{math|1=''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 {{math|''n'' ≥ 0}} and {{math|''m'' ≥ 1}}, the functions {{math|''J<sub>n</sub>''(''x'')}} and {{math|''J''<sub>''n'' + ''m''</sub>(''x'')}} have no common zeros other than the one at {{math|1=''x'' = 0}}. The hypothesis was proved by [[Carl Ludwig Siegel]] in 1929.<ref>Watson, pp. 484–485.</ref>

=== Transcendence ===
Siegel proved in 1929 that when ''ν'' is rational, all nonzero roots of {{math|''J''<sub>''ν''</sub>(x)}} and {{math|''J''{{'}}<sub>''ν''</sub>(x)}} are [[transcendental number|transcendental]],<ref name="lorch"/> as are all the roots of {{math|''K''<sub>''ν''</sub>(x)}}.<ref name="euclid"/> It is also known that all roots of the higher derivatives <math>J_\nu^{(n)}(x)</math> for {{math|''n'' ≤ 18}} are transcendental, except for the special values <math>J_1^{(3)}(\pm\sqrt3) = 0</math> and <math>J_0^{(4)}(\pm\sqrt3) = 0</math>.<ref name="lorch">{{cite journal |last1=Lorch |first1=Lee |last2=Muldoon |first2=Martin E. |title=Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |journal=International Journal of Mathematics and Mathematical Sciences |date=1995 |volume=18 |issue=3 |pages=551–560 |doi=10.1155/S0161171295000706 |doi-access=free}}</ref>

=== Numerical approaches ===
For numerical studies about the zeros of the Bessel function, see {{harvtxt|Gil|Segura|Temme|2007}}, {{harvtxt|Kravanja|Ragos|Vrahatis|Zafiropoulos|1998}} and {{harvtxt|Moler|2004}}.

=== Numerical values ===
The first zeros in J<sub>0</sub> (i.e., j<sub>0,1</sub>, j<sub>0,2</sub> and j<sub>0,3</sub>) occur at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.<ref>Abramowitz & Stegun, p409</ref>

== History ==

=== Waves and elasticity problems ===
The first appearance of a Bessel function appears in the work of [[Daniel Bernoulli]] in 1732, while working on the analysis of a [[String vibration|vibrating string]], a problem that was tackled before by his father [[Johann Bernoulli]].<ref name=":0" /> Daniel considered a flexible chain suspended from a fixed point above and free at its lower end.<ref name=":0" /> The solution of the differential equation led to the introduction of a function that is now considered  <math>J_0(x)</math>. Bernoulli also developed a method to find the zeros of the function.<ref name=":0" />

[[Leonhard Euler]] in 1736, found a link between other functions (now known as [[Laguerre polynomials]]) 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>I_n(x)</math>.<ref name=":0" /> 

In the middle of the eighteen century, [[Jean le Rond d'Alembert]] had found a [[D'Alembert's formula|formula]] to solve the [[wave equation]]. By 1771 there was dispute between Bernoulli, Euler, d'Alembert and [[Joseph-Louis Lagrange]] on the nature of the solutions vibrating strings.<ref name=":0" />

Euler worked in 1778 on [[buckling]], introducing the concept of [[Euler's critical load]]. To solve the problem he introduced the series for <math>J_{\pm 1/3}(x)</math>.<ref name=":0" /> 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>J_n(x)</math>, for integer ''n''.<ref name=":0" />

During the end of the 19th century Lagrange, [[Pierre-Simon Laplace]] and [[Marc-Antoine Parseval]] also found equivalents to the Bessel functions.<ref name=":0" /> Parseval for example found an integral representation of <math>J_0(x)</math> using cosine.<ref name=":0" />

At the beginning of the 1800s, [[Joseph Fourier]] used <math>J_0(x)</math> to solve the [[heat equation]] in a problem with cylindrical symmetry.<ref name=":0" /> Fourier won a prize of the [[French Academy of Sciences]] for this work in 1811.<ref name=":0" /> But most of the details of his work, including the use of a [[Fourier series]], remained unpublished until 1822.<ref name=":0" /> 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).<ref name=":0" />

=== Astronomical problems ===
In 1770, Lagrangre introduced the series expansion of Bessel functions to solve [[Kepler's equation]], a trascendental equation in astronomy. [[Friedrich Wilhelm Bessel]] had seen Lagrange's solution but found it difficult to handle. In 1813 in a letter to [[Carl Friedrich Gauss]], Bessel simplified the calculation using trigonometric functions.<ref name=":0" /> Bessel published his work in 1819, independently introducing the method of Fourier series unaware of the work of Fourier which was published later.<ref name=":0" />
In 1824, Bessel carried out a systematic investigation of the functions, which earned the functions his name.<ref name=":0" /> In older literature the functions were called cylindrical functions or even Bessel–Fourier functions.<ref name=":0" />

== See also ==
{{div col|colwidth=20em}}
* [[Anger function]]
* [[Bessel polynomials]]
* [[Bessel–Clifford function]]
* [[Bessel–Maitland function]]
* [[Fourier–Bessel series]]
* [[Hahn–Exton q-Bessel function|Hahn–Exton {{mvar|q}}-Bessel function]]
* [[Hankel transform]]
* [[Incomplete Bessel functions]]
* [[Jackson q-Bessel function|Jackson {{mvar|q}}-Bessel function]]
* [[Kelvin functions]]
* [[Kontorovich–Lebedev transform]]
* [[Lentz's algorithm]]
* [[Lerche–Newberger sum rule]]
* [[Lommel function]]
* [[Lommel polynomial]]
* [[Neumann polynomial]]
* [https://mathworld.wolfram.com/Riccati-BesselFunctions.html Riccati-Bessel Functions]
* [[Schlömilch's series]]
* [[Sonine formula]]
* [[Struve function]]
* [[Vibrations of a circular membrane]]
* Weber function (defined at [[Anger function]])
* [[Gauss circle problem#Exact forms|Gauss' circle problem]]
{{div col end}}

== Notes ==
{{reflist|30em}}

== References ==
{{refbegin|30em}}
* {{Abramowitz_Stegun_ref2|9|355|10|435}}
* [[George B. Arfken|Arfken, George B.]] and Hans J. Weber, ''Mathematical Methods for Physicists'', 6th edition (Harcourt: San Diego, 2005). {{ISBN|0-12-059876-0}}.
* {{cite journal |last1=Bessel |first1=Friedrich |title=Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne entsteht |journal=Berlin Abhandlungen |date=1824 |trans-title=Investigation of the part of the planetary disturbances which arise from the movement of the sun}}  Reproduced as pages 84 to 109 in {{cite book |title=Abhandlungen von Friedrich Wilhelm Bessel |date=1875 |publisher=Engelmann |location=Leipzig |url=https://play.google.com/store/books/details?id=Un4EAAAAYAAJ}} [https://drive.google.com/file/d/1W-z4BNN4s7nfGC9IbCkhQksHzGQG-wox/view?usp=sharing English translation of the text].
* Bowman, Frank ''Introduction to Bessel Functions'' (Dover: New York, 1958). {{ISBN|0-486-60462-4}}.
* {{cite book |last1=Gil |first1=A. |last2=Segura |first2=J. |last3=Temme |first3=N. M. |year=2007 |title=Numerical methods for special functions |publisher=Society for Industrial and Applied Mathematics}}
* {{citation |title=ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument |journal=Computer Physics Communications |volume=113 |issue=2–3 |pages=220–238 |year=1998 |doi=10.1016/S0010-4655(98)00064-2 |first1=P. |last1=Kravanja |first2=O. |last2=Ragos |first3=M.N. |last3=Vrahatis |first4=F.A. |last4=Zafiropoulos |bibcode=1998CoPhC.113..220K |author-link=P. Kravanja}}
* {{cite journal |last1=Mie |first1=G. |author-link=Gustav Mie |year=1908 |title=Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen |journal=Annalen der Physik |volume=25 |issue=3 |page=377 |doi=10.1002/andp.19083300302 |bibcode=1908AnP...330..377M |doi-access=free}}
* {{dlmf|first=F. W. J. |last=Olver|authorlink=Frank W. J. Olver|first2=L. C. |last2=Maximon|id=10}}.
* {{Citation |last1=Press |first1=W. H. |author-link=William H. Press |last2=Teukolsky |first2=S. A. |last3=Vetterling |first3=W. T. |last4=Flannery |first4=B. P. |year=2007 |title=Numerical Recipes: The Art of Scientific Computing |edition=3rd |publisher=Cambridge University Press |location=New York |isbn=978-0-521-88068-8 |chapter=Section 6.5. Bessel Functions of Integer Order |chapter-url=http://numerical.recipes/book/book.html |access-date=2022-09-28 |archive-date=2021-02-03 |archive-url=https://web.archive.org/web/20210203001225/http://numerical.recipes/book/book.html |url-status=dead }}.
* B Spain, M. G. Smith, ''[https://books.google.com/books?id=kYgZAQAAIAAJ&q=Bessel Functions of mathematical physics]'', Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions.
* N. M. Temme, ''Special Functions. An Introduction to the Classical Functions of Mathematical Physics'', John Wiley and Sons, Inc., New York, 1996. {{ISBN|0-471-11313-1}}. Chapter 9 deals with Bessel functions.
* [[G. N. Watson|Watson, G. N.]], ''A Treatise on the Theory of Bessel Functions, Second Edition'', (1995) Cambridge University Press. {{ISBN|0-521-48391-3}}.
* {{citation |last=Weber |first=Heinrich |author-link=Heinrich Friedrich Weber |title=Ueber eine Darstellung willkürlicher Functionen durch Bessel'sche Functionen |journal=Mathematische Annalen |volume=6 |issue=2 |pages=146–161 |year=1873 |doi=10.1007/BF01443190 |s2cid=122409461}}.
{{refend}}

== External links ==
{{refbegin|30em}}
* {{SpringerEOM|first=P. I. |last=Lizorkin|id=b/b015840|title=Bessel functions}}.
* {{SpringerEOM|first=L. N. |last=Karmazina|first2=A.P. |last2=Prudnikov|id=c/c027610|title=Cylinder function}}.
* {{SpringerEOM|first=N. Kh.|last=Rozov|id=B/b015830|title=Bessel equation}}.
* Wolfram function pages on Bessel [https://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ J] and [https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ Y] functions, and modified Bessel [https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ I] and [https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ K] functions. Pages include formulas, function evaluators, and plotting calculators.
* {{MathWorld|id=BesselFunctionoftheFirstKind|title=Bessel functions of the first kind}}
* Bessel functions [http://www.librow.com/articles/article-11/appendix-a-34 J<sub>ν</sub>], [http://www.librow.com/articles/article-11/appendix-a-35 Y<sub>ν</sub>], [http://www.librow.com/articles/article-11/appendix-a-36 I<sub>ν</sub>] and [http://www.librow.com/articles/article-11/appendix-a-37 K<sub>ν</sub>] in Librow [http://www.librow.com/articles/article-11 Function handbook].
* F. W. J. Olver, L. C. Maximon, [https://dlmf.nist.gov/10 Bessel Functions] (chapter 10 of the Digital Library of Mathematical Functions).
* {{cite book |last=Moler |first=C. B. |year=2004 |title=Numerical Computing with MATLAB |publisher=Society for Industrial and Applied Mathematics |url=http://tocs.ulb.tu-darmstadt.de/124154883.pdf |url-status=dead |archive-url=https://web.archive.org/web/20170808214249/http://tocs.ulb.tu-darmstadt.de/124154883.pdf |archive-date=2017-08-08}}
{{refend}}
{{Authority control}}

[[Category:Special hypergeometric functions]]
[[Category:Fourier analysis]]