Difference between revisions of "Manuals/calci/BESSELK"
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(Created page with "<div id="6SpaceContent" class="zcontent" align="left"> <font color="#484848"><font face="Arial, sans-serif"><font size="2">'''BESSELK'''</font></font></font><font color="#48...") |
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| − | <div | + | <div style="font-size:30px">'''BESSELK(x,n)'''</div><br/> |
| + | *Where <math>x</math> is the value at which to evaluate the function | ||
| + | *<math>n</math> 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:<math> x^2 (d^2 y/dx^2) + x(dy/dx) + (x^2 - α^2)y =0</math> | ||
| + | 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=0 to infinity){(-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== | |
| − | + | #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 | |
| − | |||
| − | + | ==See Also== | |
| + | *[[Manuals/calci/BESSELI | BESSELI ]] | ||
| + | *[[Manuals/calci/BESSELY | BESSELY ]] | ||
| + | *[[Manuals/calci/BESSELJ | BESSELJ ]] | ||
| − | + | ==References== | |
| − | + | [http://en.wikipedia.org/wiki/Absolute_value| Absolute_value] | |
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Revision as of 01:25, 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: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=0 to infinity){(-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
- 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