Bessel functions

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Explicit plots of the ${\displaystyle J_{0}}$ and ${\displaystyle J_{1}}$ from [1]
Complex map of ${\displaystyle J_{1}}$ by [2]; ${\displaystyle u+\mathrm {i} v=J_{1}(x+\mathrm {i} y)}$

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Bessel functions are solutions of the Bessel differential equation:[3][4][5]

${\displaystyle z^{2}{\frac {d^{2}w}{dz^{2}}}+z{\frac {dw}{dz}}+(z^{2}-\alpha ^{2})w=0}$

where α is a constant.

Because this is a second-order differential equation, it should have two linearly-independent solutions:

(i) Jα(x) and
(ii) Yα(x).

In addition, a linear combination of these solutions is also a solution:

(iii) Hα(x) = C1 Jα(x) + C2 Yα(x)

where C1 and C2 are constants.

These three kinds of solutions are called Bessel functions of the first kind, second kind, and third kind.

Properties

Many properties of functions $J$, $Y$ and $H$ are collected at the handbook by Abramowitz, Stegun [6].

Integral representations

${\displaystyle \!\!\!\!\!\!\!\!\!\!(9.1.20)~~~\displaystyle J_{\nu }(z)={\frac {(z/2)^{\nu }}{\pi ^{1/2}~(\nu -1/2)!}}~\int _{0}^{\pi }~\cos(z\cos(t))\sin(t)^{2\nu }~t~\mathrm {d} t}$

Expansions at small argument

${\displaystyle \displaystyle J_{\alpha }(z)=\left({\frac {z}{2}}\right)^{\!\alpha }~\sum _{k=0}^{\infty }~{\frac {(-z^{2}/4)^{k}}{k!~(\alpha \!+\!k)!}}}$

The series converges in the whole complex $z$ plane, but fails at negative integer values of ${\displaystyle \alpha }$ . The postfix form of factorial is used above; ${\displaystyle k!=\mathrm {Factorial} (k)}$.

Applications

Bessel functions arise in many applications. For example, Kepler’s Equation of Elliptical Motion, the vibrations of a membrane, and heat conduction, to name a few. In paraxial optics the Bessel functions are used to describe solutions with circular symmetry.

References

1. http://tori.ils.uec.ac.jp/TORI/index.php/File:Besselj0j1plotT.png Explicit plots of the ${\displaystyle J_{0}}$ and ${\displaystyle J_{1}}$.
2. http://tori.ils.uec.ac.jp/TORI/index.php/File:Besselj1map1T080.png Complex map of the Bessel function BesselJ1.
3. Frank Bowman (1958). Introduction to Bessel Functions, 1st Edition. Dover Publications. ISBN 0-486-60462-4.
4. George Neville Watson (1966). A Treatise on the Theory of Bessel Functions, 2nd Edition. Cambridge University Press.
5. Bessel Function of the First Kind Eric W. Weisstein, From the website of "MathWorld--A Wolfram Web Resource".
6. http://people.math.sfu.ca/~cbm/aands/page_358.htm M. Abramowitz and I. A. Stegun. Handbook of mathematical functions.