Difference between revisions of "Manuals/calci/BESSELK"
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*This function gives the value of the modified Bessel function when the arguments are purely imaginary. | *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 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:<math> x^2 | + | *Bessel's Differential Equation is defined as: |
| − | where | + | <math>x^2 \frac{d^2 y}{dx^2} + x\frac{dy}{dx} + (x^2 - \alpha^2)y =0</math> |
| + | where <math>\alpha</math> is the Arbitrary Complex number. | ||
*But in most of the cases α is the non-negative real 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. | + | *The solutions of this equation are called Bessel Functions of order <math>n</math>. |
| − | *Bessel functions of the first kind, denoted as Jn(x) | + | *Bessel functions of the first kind, denoted as <math>Jn(x)</math>. |
| − | *The Bessel function of the second kind Yn(x). | + | *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 2nd kind of order can be expressed as: Yn(x)= | + | *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). | ||
*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. | *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. | *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. | ||
Revision as of 00:45, 2 December 2013
BESSELK(x,n)
- Where is the value at which to evaluate the function
- is the integer which is the order of the Bessel Function
is the value at which to evaluate the function
is the integer which is the order of the Bessel FunctionDescription
- 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:
where is the Arbitrary Complex number.
where 
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 .
- Bessel functions of the first kind, denoted as .
- 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 .
- 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).
- 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
- BESSELK(5,2)=0.005308944 (EXCEL)Kn(x) =0.0040446134(CALCI)K1(x)
- BESSELK(0.2,4)=29900.2492 (EXCEL)Kn(x)=4.7759725484(CALCI)K1(x)
- BESSELK(10,1)=0.000155369
- BESSELK(2,-1)=NAN