Editing Bessel function (section)

=== Modified Bessel functions: ''I<sub>α</sub>'', ''K<sub>α</sub>'' <span class="anchor" id="Modified Bessel functions"></span><span class="anchor" id="Modified Bessel functions : Iα, Kα"></span> ===
The Bessel functions are valid even for [[complex number|complex]] arguments {{mvar|x}}, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the '''modified Bessel functions''' (or occasionally the '''hyperbolic Bessel functions''') '''of the first and second kind''' and are defined as<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_375.htm p. 375, 9.6.2, 9.6.10, 9.6.11].</ref>
<math display="block">\begin{align}
I_\alpha(x) &= i^{-\alpha} J_\alpha(ix) = \sum_{m=0}^\infty \frac{1}{m!\, \Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}, \\[5pt]
K_\alpha(x) &= \frac{\pi}{2} \frac{I_{-\alpha}(x) - I_\alpha(x)}{\sin \alpha \pi},
\end{align}</math>
when {{mvar|α}} is not an integer; when {{mvar|α}} is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments {{mvar|x}}. The series expansion for {{math|''I<sub>α</sub>''(''x'')}} is thus similar to that for {{math|''J<sub>α</sub>''(''x'')}}, but without the alternating {{math|(−1)<sup>''m''</sup>}} factor.

<math>K_{\alpha}</math> can be expressed in terms of Hankel functions:
<math display="block">K_{\alpha}(x) = \begin{cases}
\frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix) & -\pi < \arg x \leq \frac{\pi}{2} \\
\frac{\pi}{2} (-i)^{\alpha+1} H_\alpha^{(2)}(-ix) & -\frac{\pi}{2} < \arg x \leq \pi
\end{cases}</math>

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">
J_{\alpha}^2(x)+Y_{\alpha}^2(x)=\frac{8}{\pi^2}\int_{0}^{\infty}\cosh(2\alpha t)K_0(2x\sinh t)\, dt,
</math>

given that the condition {{math|Re(''x'') > 0}} is met. It can also be shown that
<math display="block">
J_\alpha^2(x)+Y_{\alpha}^2(x)=\frac{8\cos(\alpha\pi)}{\pi^2} \int_0^\infty K_{2\alpha}(2x\sinh t)\, dt,
</math>
only when {{math|{{abs|Re(''α'')}} < {{sfrac|1|2}}}} and {{math|Re(''x'') &ge; 0}} but not when {{math|1=''x'' = 0}}.<ref>{{cite journal |last1=Dixon |last2=Ferrar |first2=W.L. |date=1930 |title=A direct proof of Nicholson's integral |journal=The Quarterly Journal of Mathematics |location=Oxford |pages=236–238 |doi=10.1093/qmath/os-1.1.236}}</ref>

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if {{math|−''π'' < arg ''z'' ≤ {{sfrac|''π''|2}}}}):<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_375.htm p. 375, 9.6.3, 9.6.5].</ref>
<math display="block">\begin{align}
J_\alpha(iz) &= e^{\frac{\alpha\pi i}{2}} I_\alpha(z), \\[1ex]
Y_\alpha(iz) &= e^{\frac{(\alpha+1)\pi i}{2}}I_\alpha(z) - \tfrac{2}{\pi} e^{-\frac{\alpha\pi i}{2}}K_\alpha(z).
\end{align}</math>

{{math|''I<sub>α</sub>''(''x'')}} and {{math|''K<sub>α</sub>''(''x'')}} are the two linearly independent solutions to the '''modified Bessel's equation''':<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_374.htm p. 374, 9.6.1].</ref>
<math display="block">x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - \left(x^2 + \alpha^2 \right)y = 0.</math>

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, {{mvar|I<sub>α</sub>}} and {{mvar|K<sub>α</sub>}} are [[exponential growth|exponentially growing]] and [[exponential decay|decaying]] functions respectively. Like the ordinary Bessel function {{mvar|J<sub>α</sub>}}, the function {{mvar|I<sub>α</sub>}} goes to zero at {{math|1=''x'' = 0}} for {{math|''α'' > 0}} and is finite at {{math|1=''x'' = 0}} for {{math|1=''α'' = 0}}. Analogously, {{mvar|K<sub>α</sub>}} diverges at {{math|1=''x'' = 0}} with the singularity being of logarithmic type for {{mvar|K<sub>0</sub>}}, and {{math|1={{sfrac|1|2}}Γ({{abs|''α''}})(2/''x'')<sup>{{abs|''α''}}</sup>}} otherwise.<ref>{{cite book |title=Quantum Electrodynamics |last1=Greiner |first1=Walter |last2=Reinhardt |first2=Joachim |date=2009 |publisher=Springer |page=72 |isbn=978-3-540-87561-1}}</ref>

{|
| [[File:Besseli.png|none|thumb|350px|Modified Bessel functions of the first kind, <math>I_\alpha(x)</math>, for <math>\alpha = 0, 1, 2, 3</math>.]]
| [[File:Besselk.png|none|thumb|350px|Modified Bessel functions of the second kind, <math>K_\alpha(x)</math>, for <math>\alpha = 0, 1, 2, 3</math>.]]
|}
<!-- [[File:ModifiedBessel.png|Plot of some modified Bessel functions]]<br />Plot of six modified Bessel functions. In solid line {{math|''K''<sub>0</sub>}}, {{math|''K''<sub>1</sub>}}, and {{math|''K''<sub>2</sub>}}. In dashed line: {{math|''I''<sub>0</sub>}}, {{math|''I''<sub>1</sub>}}, and {{math|''I''<sub>2</sub>}}. -->

Two integral formulas for the modified Bessel functions are (for {{math|Re(''x'') > 0}}):<ref>Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA181 p. 181].</ref>
<math display="block">\begin{align}
I_\alpha(x) &= \frac{1}{\pi}\int_0^\pi e^{x\cos \theta} \cos \alpha\theta \,d\theta - \frac{\sin \alpha\pi}{\pi}\int_0^\infty e^{-x\cosh t - \alpha t} \,dt, \\[5pt]
K_\alpha(x) &= \int_0^\infty e^{-x\cosh t} \cosh \alpha t \,dt.
\end{align}</math>

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for {{math|Re(ω) > 0}}):
<math display="block">2\,K_0(\omega) = \int_{-\infty}^\infty \frac{e^{i\omega t}}{\sqrt{t^2+1}} \,dt.</math>

It can be proven by showing equality to the above integral definition for {{math|''K''<sub>0</sub>}}. This is done by integrating a closed curve in the first quadrant of the complex plane.

Modified Bessel functions of the second kind may be represented with Bassett's integral <ref>{{cite web |url=http://dlmf.nist.gov/10.32.E11
|title=Modified Bessel Functions §10.32 Integral Representations |author=<!--Not stated--> |date=<!--Not stated--> |website=NIST Digital Library of Mathematical Functions |publisher=NIST |access-date=2024-11-20}}</ref>
<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>

Modified Bessel functions {{math|''K''<sub>1/3</sub>}} and {{math|''K''<sub>2/3</sub>}} can be represented in terms of rapidly convergent integrals<ref>{{cite journal |first=M. Kh. |last=Khokonov |title=Cascade Processes of Energy Loss by Emission of Hard Photons |journal=Journal of Experimental and Theoretical Physics |volume=99 |issue=4 |pages=690–707 |date=2004 |doi=10.1134/1.1826160 |bibcode=2004JETP...99..690K |s2cid=122599440}}. Derived from formulas sourced to [[Gradshteyn and Ryzhik|I. S. Gradshteyn and I. M. Ryzhik]], ''[[Table of Integrals, Series, and Products]]'' (Fizmatgiz, Moscow, 1963; Academic Press, New York, 1980).</ref>
<math display="block"> \begin{align}
K_{\frac{1}{3}}(\xi) &= \sqrt{3} \int_0^\infty \exp \left(- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right) \,dx, \\[5pt]
K_{\frac{2}{3}}(\xi) &= \frac{1}{\sqrt{3}} \int_0^\infty \frac{3+2x^2}{\sqrt{1+\frac{x^2}{3}}} \exp \left(- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}}\right) \,dx.
\end{align}</math>

The modified Bessel function <math>K_{\frac{1}{2}}(\xi)=(2 \xi / \pi)^{-1/2}\exp(-\xi)</math> is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions.

The '''modified Bessel function of the second kind''' has also been called by the following names (now rare):
* '''Basset function''' after [[Alfred Barnard Basset]]
* '''Modified Bessel function of the third kind'''
* '''Modified Hankel function'''<ref>Referred to as such in: {{cite journal |last=Teichroew |first=D. |title=The Mixture of Normal Distributions with Different Variances |journal=The Annals of Mathematical Statistics |volume=28 |issue=2 |date=1957 |pages=510–512 |doi=10.1214/aoms/1177706981 |url=https://dml.cz/bitstream/handle/10338.dmlcz/103973/AplMat_27-1982-4_7.pdf |doi-access=free}}</ref>
* '''Macdonald function''' after [[Hector Munro Macdonald]]