Applied Mathematics/Bessel Functions

Bessel Functions

The equation below is called Bessel's differential equation.

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

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 $J_n(x)$. Bessel function is defined as follow:

$J_n(x)=\frac{x^n}{2^n\mathrm{\Gamma }(1-n)}(1-\frac{x^2}{2(2n+2)}+\frac{x^4}{2\cdot 4(2n+2)(2n+4)}-\cdots )$

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

where

$H_n^{(1)}(x)=J_n(x)+i{N}_n(x)$

$H_n^{(2)}(x)=J_n(x)-i{N}_n(x)$

$N_n(x)=\frac{J_n(x)\cos (n\pi )-J_{-n}(x)}{\sin (n\pi )}$

Γ(z) is the gamma function. $i$ is the imaginary unit. $N_n(x)$ is the Neumann function(or Bessel function of the second kind). $H_n(x)$ is the Hankel functions. If n is an integer, the Bessel function of the first kind is an entire function.