Manuals/calci/BESSELK

BESSELK(x,n)

• Where $x$ is the value at which to evaluate the function
• $n$ is the integer which is the order of the Bessel Function

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.

• 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 $J_{n}(x)$.
• The Bessel function of the first kind of order can be expressed as:

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

• 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

$K_{n}(x)=\lim _{p\to n}{\frac {\pi }{2}}{\frac {I_{-p}(x)-I_{p}(x)}{Sin(p\pi )}}$

are the modified Bessel functions of the first and second kind respectively.

• This function will give the result as error when:

1. $x$ or $n$ is non numeric 
2. $n<0$, because $n$ is the order of the function.

Examples

1. BESSELK(5,2)=0.005308944 (EXCEL)Kn(x) =0.0040446134(CALCI)K1(x)
2. BESSELK(0.2,4)=29900.2492 (EXCEL)Kn(x)=4.7759725484(CALCI)K1(x)
3. BESSELK(10,1)=0.000155369
4. BESSELK(2,-1)=NAN

See Also

• BESSELI
• BESSELY
• BESSELJ

References

Bessel Function