Bessel functions

Bessel functions are solutions of the Bessel differential equation:

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

where α is a constant.

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

(i) J_α(x) and
(ii) Y_α(x).

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

(iii) H_α(x) = 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.