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

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

$${J}_{\alpha}(x)=\sum _{m=0}^{\mathrm{\infty}}\frac{(-1{)}^{m}}{m!\Gamma (m+\alpha +1)}{\left(\frac{x}{2}\right)}^{2m+\alpha}$$`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:

$${Y}_{\alpha}(x)=\frac{{J}_{\alpha}(x)\mathrm{cos}(\alpha \pi )-{J}_{-\alpha}(x)}{\mathrm{sin}(\alpha \pi )}.$$`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 $n$:

$${Y}_{n}(x)=\underset{\alpha \to n}{\mathrm{lim}}{Y}_{\alpha}(x),$$`Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x),`

which has the result (in integral form)

$${Y}_{n}(x)=\frac{1}{\pi}{\int}_{0}^{\pi}\mathrm{sin}(x\mathrm{sin}\theta -n\theta )d\theta -\frac{1}{\pi}{\int}_{0}^{\mathrm{\infty}}[{e}^{nt}+(-1{)}^{n}{e}^{-nt}]{e}^{-x\mathrm{sinh}t}\mathrm{dt}.$$```
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.
```