Editing Bessel function (section)

=== 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>