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.