Editing Bessel function (section)

== 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>}}.