Bessel functions

Bessel functions are solutions of the Bessel differential equation:

$z²(d²w/dz²)+z(dw/dz)+(z²-alpha²)$

where alpha is a constant.

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

(i) J_alpha(x) and (ii) Y_alpha(x).

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

(iii) H_alpha = C₁ J_alpha(x) + C₂ Y_alpha(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.

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.

Bibliography

1) Weisstein, Eric W. "Bessel Function of the First Kind." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

2) “Introduction to Bessel Functions” by Frank Bowman Dover Publications, Inc. NewYork 1958

3) “A Treatise on the Theory of Bessel Functions” by G. N. Watson Second Edition Cambridge University Press 1966