Difference between revisions of "Manuals/calci/BESSELK"

Revision as of 00:46, 2 December 2013

BESSELK(x,n)

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.

But in most of the cases α is the non-negative real number.
The solutions of this equation are called Bessel Functions of order $n$ .
Bessel functions of the first kind, denoted as $Jn(x)$ .
The Bessel function of the first kind of order can be expressed as:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Jn(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 $Yn(x)$ .
The Bessel function of the 2nd kind of order can be expressed as: $Yn(x)=\lim _{p\to n}{J_{p}(x)Cos(p\pi )-J_{-p}(x)}/Sin(p\pi )$ .
So the form of the general solution is y(x)=c1 In(x)+c2 Kn(x). where In(x)=i^-nJn(ix) and Kn(x)=lt p tends to n pi()/2[( I-p(x)-I p(x))/Sinp pi()] are the modified Bessel functions of the first and second kind respectively.
This function will give the result as error when 1.x or n is non numeric 2. n<0, because n is the order of the function.

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 *The Bessel function of the first kind of order can be expressed as:<math>Jn(x)=sum_{k=0}^\infty}\frac{(-1)^k}{k!\Gamma(n+k+1)}.(\frac{x}{2})^{n+2k}</math>
 *The Bessel function of the second kind  <math>Yn(x)</math>.
-*The Bessel function of the 2nd kind of order  can be expressed as: <math>Yn(x)=\lim_{p \to n}{J_p(x)Cos(p\pi)- J_{-p}(x)}/Sin(p\pi).
+*The Bessel function of the 2nd kind of order  can be expressed as: <math>Yn(x)=\lim_{p \to n}{J_p(x)Cos(p\pi)- J_{-p}(x)}/Sin(p\pi)</math>.
 *So the form of the general solution is y(x)=c1 In(x)+c2 Kn(x). where In(x)=i^-nJn(ix) and Kn(x)=lt p tends to n  pi()/2[( I-p(x)-I p(x))/Sinp pi()] are the modified Bessel functions of the first and second kind respectively.
 *This function will give the result as error when 1.x or n is non numeric 2. n<0, because n is the order of the function.