Fortran Wiki
Bessel function

Bessel functions, are canonical solutions y(x) of Bessel’s differential equation:

x 2d 2ydx 2+xdydx+(x 2α 2)y=0x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2) y = 0

for an arbitrary real or complex number α (the order of the Bessel function). The most common and important special case is where α is an integer (in which case we call it n).

Bessel Functions of the First Kind: J α

Bessel functions of the first kind, denoted J α(x), are solutions of Bessel’s differential equation that are finite at the origin (x=0) for non-negative integer α, and diverge as x0 for negative non-integer α. It is possible to define the function by its Taylor series expansion around x=0:

J α(x)= m=0 (1) mm!Γ(m+α+1)(x2) 2m+αJ_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \Gamma(m+\alpha+1)} {\left({\frac{x}{2}}\right)}^{2m+\alpha}

where Γ(z) is the gamma function, a generalization of the factorial function to non-integer values.

Bessel Functions of the Second Kind: Y α

Bessel functions of the second kind, denoted by Y α(x), are solutions of the Bessel differential equation. They are singular (infinite) at the origin (x=0).

For non-integer α, it is related to J α(x) by:

Y α(x)=J α(x)cos(απ)J α(x)sin(απ).Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}.

In the case of integer order n, the function is defined by taking the limit as a non-integer α tends to n:

Y n(x)=lim αnY α(x),Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x),

which has the result (in integral form)

Y n(x)=1π 0 πsin(xsinθnθ)dθ1π 0 [e nt+(1) ne nt]e xsinhtdt.Y_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \sin(x \sin\theta - n\theta)d\theta - \frac{1}{\pi} \int_{0}^{\infty} \left[ e^{n t} + (-1)^n e^{-n t} \right] e^{-x \sinh t} dt.