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# Bessel functions

**Bessel functions** are solutions of the Bessel differential equation:^{[3]}^{[4]}^{[5]}

where α is a constant.

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

(i) J_{α}(x) and

(ii) Y_{α}(x).

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

(iii) H_{α}(x) = C_{1} J_{α}(x) + C_{2} Y_{α}(x)

where C_{1} and C_{2} are constants.

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

## Contents

## Properties

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

### Integral representations

### Expansions at small argument

The series converges in the whole complex $z$ plane, but fails at negative integer values of . The postfix form of factorial is used above; .

## 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

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