**Bessel functions**, are canonical solutions $y(x)$ of Bessel’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 $n$).

## 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 $x = 0$:

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

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

## 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:

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

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

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

which has the result (in integral form)

$Y_n(x) =
\frac{1}{\pi} \int_{0}^{\pi} \sin(x \sin\theta - n\theta)d\theta
- \frac{1}{\pi} \int_{0}^{\infty}
\left[ e^{n t} + (-1)^n e^{-n t} \right]
e^{-x \sinh t} dt.$