Mathematical Methods for Physicists: A concise introduction - Site Map

7
Special functions of
mathematical physics
The functions discussed in this chapter arise as solutions of second-order di€erential
equations which appear in special, rather than in general, physical problems.
So these functions are usually known as the special functions of
mathematical physics. We start with Legendre's equation (Adrien Marie
Legendre, 1752±1833, French mathematician).
Legendre's di€erential equation
Legendre's equation
…1 x 2 † d2 y dy
2x ‡ … ‡ 1†y ˆ 0;
dx2 dx …7:1†
where v is a positive constant, is of great importance in classical and quantum
physics. The reader will see this equation in the study of central force motion in
quantum mechanics. In general, Legendre's equation appears in problems in
classical mechanics, electromagnetic theory, heat, and quantum mechanics, with
spherical symmetry.
Dividing Eq. (7.1) by 1 x 2 , we obtain the standard form
d 2 y
dx 2 2x dy
1 x 2 dx
‡
… ‡ 1†
1 x 2 y ˆ 0:
We see that the coecients of the resulting equation are analytic at x ˆ 0, so the
origin is an ordinary point and we may write the series solution in the form
y ˆ X1
mˆ0
a m x m :
…7:2†
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