# Applied Mathematics/Bessel Functions

## Bessel Functions

The equation below is called Bessel's differential equation.

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-n^{2})y=0}$

The two distinctive solutions of Bessel's differential equation are either one of the two pairs: (1)Linear combination of Bessel function(also known as Bessel function of the first kind) and Neumann function(also known as Bessel function of the second kind) (2)Linear combination of Hankel function of the first kind and Hankel function of the second kind. Bessel function (of the first kind) is denoted as ${\displaystyle \displaystyle J_{n}(x)}$ . Bessel function is defined as follow:

${\displaystyle J_{n}(x)={\frac {x^{n}}{2^{n}\Gamma (1-n)}}(1-{\frac {x^{2}}{2(2n+2)}}+{\frac {x^{4}}{2\cdot 4(2n+2)(2n+4)}}-\cdots )}$
${\displaystyle =\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+n+1)}}{\left({\frac {x}{2}}\right)}^{2m+n}}$

where

${\displaystyle H_{n}^{(1)}(x)=J_{n}(x)+iN_{n}(x)}$
${\displaystyle H_{n}^{(2)}(x)=J_{n}(x)-iN_{n}(x)}$
${\displaystyle N_{n}(x)={\frac {J_{n}(x)\cos(n\pi )-J_{-n}(x)}{\sin(n\pi )}}}$

Γ(z) is the gamma function. ${\displaystyle i}$  is the imaginary unit. ${\displaystyle N_{n}(x)}$  is the Neumann function(or Bessel function of the second kind). ${\displaystyle H_{n}(x)}$  is the Hankel functions. If n is an integer, the Bessel function of the first kind is an entire function.