Editing Bessel function (section)

{{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]].