Editing Bessel function (section)

== 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>