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:Failed to parse (syntax error): {\displaystyle x^2 (d^2 y/dx^2) + x(dy/dx) + (x^2 - α^2)y =0}

where α is the arbitary 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), and The Bessel function of the first kind of order can be expressed as:$Jn(x)=summation(k=0toinfinity){(-1)^{k}(x/2)^{n}+2k}/k!gamma(n+k+1)$.
• The Bessel function of the second kind Yn(x).
• The Bessel function of the 2nd kind of order can be expressed as: Yn(x)=lt p tends to n {Jp(x)Cosp pi()- J-p(x)}/Sinp 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.

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

Absolute_value