Bessel functions

Bessel functions are solutions of the Bessel differential equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^2 \frac {d^2 w}{dz^2} + z \frac {dw}{dz} + (z^2 - α ^2) }

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 = C₁ J_α(x) + C₂ Y_α(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

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

“Introduction to Bessel Functions”
by Frank Bowman
Dover Publications, Inc.
New York
1958

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