Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
where s n …x† is given by
s n …x† ˆXn
kˆ0
u k …x†v nk …x† ˆu 0 …x†v n …x†‡‡u n …x†v 0 …x†:
Now the Cauchy product for our two series is given by
X 1
X n
nˆ0 kˆ0
a nk z nk e …nk†i
a k z k e ki ˆ X1
nˆ0
z n Xn
kˆ0
a k a nk e …n2k†i : …7:18†
In the inner sum, which is the sum of interest to us, it is straightforward to prove
that, for n 1, the terms corresponding to k ˆ j and k ˆ n j are identical except
that the exponents on e are of opposite sign. Hence these terms can be paired, and
we have for the coecient of z n ,
P n …cos † ˆa 0 a n …e ni ‡ e ni †‡a 1 a n1 …e …n2†i ‡ e …n2†i †‡
ˆ 2‰ a 0 a n cos n ‡ a 1 a n1 cos…n 2† ‡Š:
…7:19†
If n is odd, the number of terms is even and each has a place in one of the pairs. In
this case, the last term in the sum is
a …n1†=2 a …n‡1†=2 cos :
If n is even, the number of terms is odd and the middle term is unpaired. In this
case, the series (7.19) for P n …cos † ends with the constant term
a n=2 a n=2 :
Using Eq. (7.17) to compute values of the a n , we ®nd from the unit coecient of z 0
in Eqs. (7.18) and (7.19), whether n is odd or even, the speci®c expressions
9
P 0 …cos † ˆ1; P 1 …cos † ˆcos ; P 2 …cos † ˆ…3 cos 2 ‡ 1†=4
P 3 …cos † ˆ…5 cos 3 ‡ 3 cos †=8
P 4 …cos † ˆ…35 cos 4 ‡ 20 cos 2 ‡ 9†=64
P 5 …cos † ˆ…63 cos 5 ‡ 35 cos 3 ‡ 30 cos †=128
P 6 …cos † ˆ…231 cos 6 ‡ 126 cos 4 ‡ 105 cos 2 ‡ 50†=512
>=
: …7:20†
>;
Orthogonality of Legendre polynomials
The set of Legendre polynomials fP n …x†g is orthogonal for 1 x ‡1. In
particular we can show that
Z ‡1
2=…2n ‡ 1† if m ˆ n
P n …x†P m …x†dx ˆ
0 if m 6ˆ n : …7:21†
1
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