Bessel functions: Difference between revisions
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Because this is a second-order differential equation, it should have two linearly-independent solutions: | Because this is a second-order differential equation, it should have two linearly-independent solutions: | ||
(i) J<sub>α</sub>(x) and | (i) J<sub>α</sub>(x) and<br/> | ||
(ii) Y<sub>α</sub>(x). | (ii) Y<sub>α</sub>(x). | ||
Revision as of 15:16, 6 September 2010
Bessel functions are solutions of 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 ''z''<sup>2</sup> (d<sup>2</sup>w/dz<sup>2</sup>) + z (dw/dz) + (z<sup>2</sup> - α<sup>2</sup>)}
where α is a constant.
Because this is a second-order differential equation, it should have two linearly-independent solutions:
(i) Jα(x) and
(ii) Yα(x).
In addition, a linear combination of these solutions is also a solution:
(iii) Hα = C1 Jα(x) + C2 Yα(x)
where C1 and C2 are constants.
These three kinds of solutions are called Bessel functions of the first kind, second kind, and third kind.
Applications
Bessel functions arise in many applications. For example, Kepler’s Equation of Elliptical Motion, the vibrations of a membrane, and heat conduction, to name a few.
Bibliography
1) Weisstein, Eric W. "Bessel Function of the First Kind." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
2) “Introduction to Bessel Functions” by Frank Bowman Dover Publications, Inc. NewYork 1958
3) “A Treatise on the Theory of Bessel Functions” by G. N. Watson Second Edition Cambridge University Press 1966