Difference between revisions of "Manuals/calci/BESSELK"

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*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 <math>n</math>.  
 
*The solutions of this equation are called Bessel Functions of order <math>n</math>.  
−
*Bessel functions of the first kind, denoted as <math>Jn(x)</math>.  
+
*Bessel functions of the first kind, denoted as <math>J_n(x)</math>.  
 
*The Bessel function of the first kind of order can be expressed as:
 
*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>
+
:<math>J_n(x)=\sum_{k=0}^\infty \frac{(-1)^k}{k!\Gamma(n+k+1)}.(\frac{x}{2})^{n+2k}</math>
  
−
*The Bessel function of the second kind  <math>Yn(x)</math>.
+
*The Bessel function of the second kind  <math>Y_n(x)</math>.
−
*The Bessel function of the 2nd kind of order  can be expressed as: <math>Yn(x)= \lim_{p \to n}\frac{J_p(x)Cos(p\pi)- J_{-p}(x)}{Sin(p\pi)}</math>
+
*The Bessel function of the 2nd kind of order  can be expressed as: <math>Y_n(x)= \lim_{p \to n}\frac{J_p(x)Cos(p\pi)- J_{-p}(x)}{Sin(p\pi)}</math>
−
*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 <math>y(x)=c1 I_n(x)+c2 K_n(x)</math>. where <math>I_n(x)=i^{-n}J_n(ix)</math> and <math>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:
 
*This function will give the result as error when:
 
  1. <math>x</math> or <math>n</math> is non numeric  
 
  1. <math>x</math> or <math>n</math> is non numeric  

Revision as of 01:55, 3 December 2013

BESSELK(x,n)


  • Where 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 x} is the value at which to evaluate the function
  • 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 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 (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 x^2 \frac{d^2 y}{dx^2} + x\frac{dy}{dx} + (x^2 - \alpha^2)y =0} 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 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 n} .
  • Bessel functions of the first kind, denoted 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 J_n(x)} .
  • 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 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 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 Y_n(x)} .
  • The Bessel function of the 2nd 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 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 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 y(x)=c1 I_n(x)+c2 K_n(x)} . where 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 I_n(x)=i^{-n}J_n(ix)} and 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 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. <math>x} or 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 n} is non numeric
2. 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 n<0}
, because 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 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

References

Bessel Function

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