Bessel functions: Difference between revisions
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imported>Milton Beychok m (Reformatted "Bibliography" as "References". See Talk page for rationale.) |
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Bessel functions are solutions of the Bessel differential equation: | '''Bessel functions''' are solutions of the Bessel differential equation:<ref>{{cite book|author=Frank Bowman|title=Introduction to Bessel Functions|edition=1st Edition|publisher=Dover Publuications|year=1958|id=ISBN 0-486-60462-4}}</ref><ref>{{cite book|author=George Neville Watson|title=A Treatise on the Theory of Bessel Functions|edition=2nd Edition|publisher=Cambridge University Press|year=1966|id=}}</ref><ref>[http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html Bessel Function of the First Kind] Eric W. Weisstein, From the website of "MathWorld--A Wolfram Web Resource".</ref> | ||
:<math> z^2 \frac {d^2 w}{dz^2} + z \frac {dw}{dz} + (z^2 - \alpha^2)w = 0 </math> | :<math> z^2 \frac {d^2 w}{dz^2} + z \frac {dw}{dz} + (z^2 - \alpha^2)w = 0 </math> | ||
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These three kinds of solutions are called Bessel functions of the first kind, second kind, and third kind. | 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, [[Johannes Kepler|Kepler]]’s [[Kepler's laws|Equation of Elliptical Motion]], the vibrations of a membrane, and heat conduction, to name a few. | Bessel functions arise in many applications. For example, [[Johannes Kepler|Kepler]]’s [[Kepler's laws|Equation of Elliptical Motion]], the vibrations of a membrane, and heat conduction, to name a few. | ||
== | ==References== | ||
{{reflist}} | |||
Revision as of 20:26, 7 September 2010
Bessel functions are solutions of the Bessel differential equation:[1][2][3]
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.
References
- ↑ Frank Bowman (1958). Introduction to Bessel Functions, 1st Edition. Dover Publuications. ISBN 0-486-60462-4.
- ↑ George Neville Watson (1966). A Treatise on the Theory of Bessel Functions, 2nd Edition. Cambridge University Press.
- ↑ Bessel Function of the First Kind Eric W. Weisstein, From the website of "MathWorld--A Wolfram Web Resource".