Manuals/calci/BESSELK

BESSELK(x,n)

Description

This function gives the value of the modified Bessel function when the arguments are purely imaginary.
Bessel functions is also called cylinder functions because they appear in the solution to Laplace's equation in cylindrical coordinates.
Bessel's Differential Equation is defined as:

$x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-\alpha ^{2})y=0$ where $\alpha$ is the Arbitrary Complex number.

J_{n}(x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!\Gamma (n+k+1)}}.({\frac {x}{2}})^{n+2k}

The Bessel function of the second kind $Y_{n}(x)$ .
The Bessel function of the 2nd kind of order can be expressed as: $Y_{n}(x)=\lim _{p\to n}{\frac {J_{p}(x)Cos(p\pi )-J_{-p}(x)}{Sin(p\pi )}}$
So the form of the general solution is $y(x)=c1I_{n}(x)+c2K_{n}(x)$ . where $I_{n}(x)=i^{-n}J_{n}(ix)$ and Failed to parse (syntax error): {\displaystyle K_n(x)=\lim_{p \to n}\frac{\pi}{2}\frac{ I-p(x)-I p(x)}{Sin(p\pi}} are the modified Bessel functions of the first and second kind respectively. *This function will give the result as error when: 1. <math>x} or $n$ is non numeric

2.  $n<0$ , because  $n$  is the order of the function.