Bessel functions: Difference between revisions
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Carl McBride (talk | contribs) (New page: '''Bessel functions''' of the first kind <math>J_n(x)</math> are defined as the solutions to the Bessel differential equation :<math>x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2-n^2)y=0...) |
Carl McBride (talk | contribs) m (Added applications section.) |
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:<math>J_n (z) = \frac{1}{2 \pi i} \oint e^{(z/2)(t-1/t)}t^{-n-1}{\rm d}t</math> | :<math>J_n (z) = \frac{1}{2 \pi i} \oint e^{(z/2)(t-1/t)}t^{-n-1}{\rm d}t</math> | ||
==Applications in statistical mechanics== | |||
*[[Computational implementation of integral equations]] | |||
==See also== | ==See also== | ||
*[http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html Bessel Function of the First Kind -- from Wolfram MathWorld] | *[http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html Bessel Function of the First Kind -- from Wolfram MathWorld] | ||
[[category: mathematics]] | [[category: mathematics]] | ||
Latest revision as of 10:58, 7 July 2008
Bessel functions of the first kind Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n(x)} are defined as the solutions to the Bessel differential equation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2-n^2)y=0}
which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The Bessel function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n(z)} can also be defined by the contour integral
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n (z) = \frac{1}{2 \pi i} \oint e^{(z/2)(t-1/t)}t^{-n-1}{\rm d}t}