Editing Bessel function (section)

== 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]]).