Difference between revisions of "Manuals/calci/BESSELK"
Jump to navigation
Jump to search
| Line 7: | Line 7: | ||
*Bessel's Differential Equation is defined as: | *Bessel's Differential Equation is defined as: | ||
<math>x^2 \frac{d^2 y}{dx^2} + x\frac{dy}{dx} + (x^2 - \alpha^2)y =0</math> | <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 | + | 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 <math>n</math>. | *The solutions of this equation are called Bessel Functions of order <math>n</math>. | ||
Revision as of 03:32, 5 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 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 \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 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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. 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}
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
- BESSELK(5,2) = 0.0040446134
- BESSELK(0.2,4) = 29900.2492
- BESSELK(10,1) = 0.000155369
- BESSELK(2,-1) = NAN