Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
Figure 7.1.
Legendre polynomials.
If we write D n u as u n and D n v as v n , then
…uv† n ˆ uv n ‡ n C 1 u 1 v n1 ‡‡ n C r u r v nr ‡‡u n v;
where D ˆ d=dx and n C r is the binomial coecient and is equal to n!=‰r!…n r†!Š.
We ®rst notice that Eq. (7.12) holds for n ˆ 0, 1. Then, write
so that
z ˆ…x 2 1† n =2 n n!
…x 2 1†Dz ˆ 2nxz:
Di€erentiating Eq. (7.13) …n ‡ 1† times by the Leibnitz rule, we get
…7:13†
Writing y ˆ D n z, we then have:
…1 x 2 †D n‡2 z 2xD n‡1 z ‡ n…n ‡ 1†D n z ˆ 0:
(i) y is a polynomial.
(ii) The coecient of x n in …x 2 1† n is …1† n=2 n C n=2 (n even) or 0 (n odd).
Therefore the lowest power of x in y…x† is x 0 (n even) or x 1 (n odd). It
follows that
and
y n …0† ˆ0
…n odd†
y n …0† ˆ 1
2 n n! …1†n=2 n C n=2 n! ˆ …1†n=2 n!
2 n ‰…n=2†!Š 2 …n even†:
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