Difference between revisions of "Manuals/calci/BESSELJ"
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*The solutions of this equation are called Bessel Functions of order n. | *The solutions of this equation are called Bessel Functions of order n. | ||
*Bessel functions of the first kind, denoted as <math>Jn(x)</math> | *Bessel functions of the first kind, denoted as <math>Jn(x)</math> | ||
| − | *The Bessel function of the first kind of order can be expressed as: <math>Jn(x)=\sum_{k=0}^\ | + | *The Bessel function of the first kind of order can be expressed as: |
| − | *This function will give | + | <math>Jn(x)=\sum_{k=0}^\infty \frac{(-1)^k}{k!\Gamma(n+k+1)}.(\frac{x}{2})^{n+2k}</math> |
| + | *where <math>\Gamma(n+k+1)=(n+k)!<math> or | ||
| + | *\int\limits_{0}^{\infty} x^{n+k}*e^{-x} dx is the gamma function. | ||
| + | *This function will give result as error when | ||
| + | 1. <math>x</math> or <math>n</math> is non numeric | ||
| + | 2. <math>n<0</math>, because <math>n</math> is the order of the function | ||
==Examples== | ==Examples== | ||
Revision as of 22:32, 1 December 2013
BESSELJ(x,n)
- 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 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 order of the Bessel function and is an integer
Description
- This function gives the value of the modified Bessel function.
- 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 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 non-negative real number.
- The solutions of this equation are called Bessel Functions of order 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 Jn(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 Jn(x)=\sum_{k=0}^\infty \frac{(-1)^k}{k!\Gamma(n+k+1)}.(\frac{x}{2})^{n+2k}}
- 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 \Gamma(n+k+1)=(n+k)!<math> or *\int\limits_{0}^{\infty} x^{n+k}*e^{-x} dx is the gamma function. *This function will give 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
- BESSELJ(2,3)=0.12894325(EXCEL)Jn(x)=0.10728467204(calci)J1(x)0.5767248079(Actual)J1(x)
- BESSELJ(7,2)=-0.301417224(EXCEL)Jn(x)=NAN(calci)=-0.0046828257(Actual)J1(x)
- BESSELJ(5,1)=-0.327579139(EXCEL)Jn(x)=NAN(calci)