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Date: Mon, 12 Jan 2004 16:59:37 -0600
From: Ed Pegg Jr
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To: Ed Pegg Jr
Subject: [Fwd: Re: Common uses of special functions]
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-------- Original Message --------
Subject: Re: Common uses of special functions
Date: Thu, 08 Jan 2004 12:48:28 -0600
From: Michael Trott
To: Ben Wilson
CC: edp@wolfram.com
References: <028801c3d616$40bd53c0$7304b18c@wri.wolfram.com>
>Is it possible to say what the most important or most influential special
>functions are (top 5)?... Besides the elementary functions....
Yes it is, but it wouldn't be useful.
>For example--and I'm guessing at these, so if you have better examples,
>please add or replace what I have:
Here are some more:
- Elliptic integrals arise from simple electrostatic and magnetostatic problems
- Legendre functions arise in problems having toroidal symmetry and in the description of
a penny rolling on a table
- Hermite polynomials arise in the quantum mechanical version of the harmonic oscillator
- Chebyshev polynomials appear everywhere in numerical analysis
- Mathieu functions appear in the description of ellipse-shaped drums
- Polylogarithmic functions appear everywhere in the descriptions of elementary particles
Basically for every special function there is a more general one
where the first one is a special case of.
And the most general one is the generialized Meijer G function.
While Sin has 1 argument, Bessels have two, Meijer G has as many
as you want, so you can basically build as complicated functions as you want
(not quite true, but roughly).
If you really wanted to list some important ones:
- error functions
- Bessel functions
- elliptic integrals
- hypergeometric functions
- orthogonal polynomials