Applied Mathematics/Bessel Functions

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Bessel Functions[edit]

The equation below is called Bessel's differential equation.

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

 J_n(x) = \frac{x^n}{2^n \Gamma(1-n)} (1- \frac{x^2}{2(2n+2)}+\frac{x^4}{2\cdot4(2n+2)(2n+4)}-\cdots)
 = \sum_{m=0}^\infty \frac{(-1)^m}{m! \, \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.