Bessel function: Difference between revisions

{{Short description|Families of solutions to related differential equations}}

{{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]].

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:

* 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]])

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]]).

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.

<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>

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]].

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''^{''i'' ''f''(x)}}}. 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]].

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"> 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>

[[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_n''(''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_n''(''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]]

j_n(x) &= \sqrt{\frac{\pi}{2x}}J_{n+\frac{1}{2}}(x) = \\

&= \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]

-\dfrac{\Gamma(\alpha)}{\pi} \left( \dfrac{2}{z} \right)^\alpha + \dfrac{1}{\Gamma(\alpha+1)} \left(\dfrac{z}{2} \right)^\alpha \cot(\alpha \pi) & \text{if } \alpha \text{ is a positive integer (one term dominates unless } \alpha \text{ is imaginary)}, \\[1ex]

where {{mvar|γ}} is the [[Euler–Mascheroni constant]] (0.5772...).

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, {{math|''J_α''(''z'')}} is not asymptotic to the average of these two asymptotic forms when {{mvar|z}} is negative (because one or the other will not be correct there, depending on the {{math|arg ''z''}} used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for ''complex'' (non-real) {{mvar|z}} so long as {{math|{{abs|''z''}}}} goes to infinity at a constant phase angle {{math|arg ''z''}} (using the square root having positive real part):

For integer order {{math|1=''α'' = ''n''}}, {{mvar|J_n}} is often defined via a [[Laurent series]] for a generating function:

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 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}

for {{math|''α'' > −{{sfrac|1|2}}}}. The Hankel transform can express any square-integrable function defined on <math>[0, \infty)</math> as an integral of Bessel functions of different scales.<ref>{{cite book |title=Hilbert spaces of entire functions |url=https://archive.org/details/hilbertspacesofe0000debr |url-access=registration |year=1968 |publisher=Prentice-Hall |location=London |isbn=978-0133889000 |author=Louis de Branges |authorlink=Louis de Branges de Bourcia |page=[https://archive.org/details/hilbertspacesofe0000debr/page/189 189]}}</ref> For the spherical Bessel functions the orthogonality relation is:

[[Leonhard Euler]] in 1736, found a link between other functions (now known as [[Laguerre polynomials]]) and Bernoulli's solution. Euler also introduced a non-uniform chain that lead to the introduction of functions now related to modified Bessel functions <math>I_n(x)</math>.<ref name=":0" />  

During the end of the 19th century Lagrange, [[Pierre-Simon Laplace]] and [[Marc-Antoine Parseval]] also found equivalents to the Bessel functions.<ref name=":0" /> Parseval for example found an integral representation of <math>J_0(x)</math> using cosine.<ref name=":0" />

At the beginning of the 1800s, [[Joseph Fourier]] used <math>J_0(x)</math> to solve the [[heat equation]] in a problem with cylindrical symmetry.<ref name=":0" /> Fourier won a prize of the [[French Academy of Sciences]] for this work in 1811.<ref name=":0" /> But most of the details of his work, including the use of a [[Fourier series]], remained unpublished until 1822.<ref name=":0" /> Poisson in rivalry with Fourier, extended Fourier's work in 1823, introducing new properties of Bessel functions including Bessel functions of half-integer order (now known as spherical Bessel functions).<ref name=":0" />

In 1770, Lagrange introduced the series expansion of Bessel functions to solve [[Kepler's equation]], a transcendental equation in astronomy. [[Friedrich Wilhelm Bessel]] had seen Lagrange's solution but found it difficult to handle. In 1813 in a letter to [[Carl Friedrich Gauss]], Bessel simplified the calculation using trigonometric functions.<ref name=":0" /> Bessel published his work in 1819, independently introducing the method of Fourier series unaware of the work of Fourier which was published later.<ref name=":0" />