Bessel function

**Bessel functions**, are canonical solutions $Y_n(x) = \\
Y_n(x) =
\\y(x)$ of Besselâ€™s differential equation:

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

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 $x^2 \\\\\\n$).

`J_\\alpha`

Bessel functions of the first kind, denoted `J_\\alpha(x)`

, are solutions of Besselâ€™s differential equation that are finite at the origin ($J_\\J_\\x = 0$) 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 $\\x \\\\x = 0$:

```
J_\\alpha(x) = \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\Gamma(m+\\alpha+1)} {\\left({\\frac{x}{2}}\\right)}^{2m+\\alpha}
```

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.

`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 ($J_\\\\Y_\\Y_\\x = 0$).

For non-integer `\\alpha`

, it is related to `J_\\alpha(x)`

by:

```
Y_\\alpha(x) = \\frac{J_\\alpha(x) \\cos(\\alpha\\pi) - J_{-\\alpha}(x)}{\\sin(\\alpha\\pi)}.
```

In the case of integer order $\\J_\\
Y_\\n$, the function is defined by taking the limit as a non-integer `\\alpha`

tends to $\\n$:

```
Y_n(x) = \\lim_{\\alpha \\to n} Y_\\alpha(x),
```

which has the result (in integral form)

```
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.
```

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