Manuals/calci/BESSELI

BESSELI(x,n)

• $x$ is the value to evaluate the function
• $n$ is an integer which is the order of the Bessel function

Description

• This function gives the value of the modified Bessel function.
• 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 $n^{th}$ order modified Bessel function of the variable $x$ is:

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

• 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. BESSELI(3,2) = 2.245212431 this is the $2^{nd}$ derivative of (I_n(x)).
2. BESSELI(5,1) = 24.33564185
3. BESSELI(6,0) = 67.23440724
4. BESSELI(-2,1) = 0.688948449
5. BESSELI(2,-1) = NAN ,because n<0.

See Also

• BESSELJ
• BESSELK
• BESSELY

References

Bessel Function