where α is a constant.
Because this is a second-order differential equation, it should have two linearly-independent solutions:
(i) Jα(x) and
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.
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; .
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.
- 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.