Fortran Wiki
Bessel function

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

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

Bessel Functions of the First Kind: $J_\alpha$

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

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

For evaluating Bessel functions of the first kind in Fortran, see bessel_j0, bessel_j1, and bessel_jn.

Bessel Functions of the Second Kind: $Y_\alpha$

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

For non-integer $\alpha$, it is related to $J_\alpha(x)$ by:

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

which has the result (in integral form)

For evaluating Bessel functions of the second kind in Fortran, see bessel_y0, bessel_y1, and bessel_yn.