Bessel functions: Difference between revisions
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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. | ||
==Properties== | |||
Many properties of functions $J$, $Y$ and $H$ are collected at the handbook by [[Abramowitz, Stegun]] | |||
<ref> | |||
http://people.math.sfu.ca/~cbm/aands/page_358.htm | |||
M. Abramowitz and I. A. Stegun. | |||
Handbook of mathematical functions. | |||
</ref>. | |||
===Integral representations=== | |||
===Expansions at small argument=== | |||
: <math>\displaystyle J_\alpha(z) | |||
=\left(\frac{z}{2}\right)^{\!\alpha} ~ | |||
\sum_{k=0}^{\infty} | |||
~ \frac{ (-z^2/4)^k}{ k! ~ (\alpha\!+\!k)!} | |||
</math> | |||
The series converges in the whole complex plane, but fails at negative integer values of <math>\alpha</math> . The postfix form of [[factorial]] is used above; <math>k!</math>=\mathrm{Factorial}(k)</math>. | |||
==Applications== | ==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. | ||
In [[paraxial optics]] the Bessel functions are used to describe solutions with circular symmetry. | |||
==References== | ==References== | ||
{{reflist}} | {{reflist}} | ||
Revision as of 04:52, 13 July 2012
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(x) = 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.
Properties
Many properties of functions $J$, $Y$ and $H$ are collected at the handbook by Abramowitz, Stegun [4].
Integral representations
Expansions at small argument
The series converges in the whole complex plane, but fails at negative integer values of . The postfix form of factorial is used above; =\mathrm{Factorial}(k)</math>.
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. In paraxial optics the Bessel functions are used to describe solutions with circular symmetry.
References
- ↑ Frank Bowman (1958). Introduction to Bessel Functions, 1st Edition. Dover Publications. 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".
- ↑ http://people.math.sfu.ca/~cbm/aands/page_358.htm M. Abramowitz and I. A. Stegun. Handbook of mathematical functions.