Page 789 - Handbook Of Integral Equations

0
0
The functions P ν (z) ≡ P (z) and Q ν (z) ≡ Q (z) are called the Legendre functions.
ν
ν
The modified associated Legendre functions, on the cut z = x, –1< x < 1, of the real axis are
defined by the formulas
1
µ
µ
µ

P (x)= 1 2 e 2 1 iµπ P (x + i0) + e – iµπ P (x – i0) ,
2
ν
ν
ν
1
µ
µ
1 –iµπ
µ

Q (x)= e e – iµπ Q (x + i0) + e 2 1 iµπ Q (x – i0) .
2
ν
ν
ν
2
Trigonometric expansions
For –1< x < 1, the modified associated Legendre functions can be represented in the form
trigonometric series:
1
∞
2 µ+1 Γ(ν + µ +1) ( + µ) k (1 + ν + µ) k
µ µ 2
P (cos θ)= √ (sin θ) sin[(2k + ν + µ +1)θ],
ν
3
3
π Γ(ν + ) k!(ν + ) k
2 k=0 2
∞ 1
√ µ Γ(ν + µ +1) µ ( + µ) k (1 + ν + µ) k
µ
Q (cos θ)= π 2 (sin θ) 2 cos[(2k + ν + µ +1)θ],
ν 3 3
Γ(ν + ) k!(ν + ) k
2 k=0 2
where 0 < θ < π.
Some relations
Γ(ν + n +1) –n
µ
n
µ
P (z)= P (z), P (z)= P (z), n =0, 1, 2, ...
ν –ν–1 ν ν
Γ(ν – n +1)
π iπµ µ Γ(1 + ν + µ) –µ
µ
Q (z)= e P (z) – P (z) .
ν ν ν
2 sin(µπ) Γ(1 + ν – µ)
For 0 < x <1,
µ
–1
µ
µ
P (–x)= P (x) cos[π(ν + µ)] – 2π Q (x) sin[π(ν + µ)],
ν
ν
ν
µ
1
µ
µ
Q (–x)= –Q (x) cos[π(ν + µ)] – πP (x) sin[π(ν + µ)].
ν
ν
ν
2
For –1< x <1,
µ 2ν +1 µ ν + µ µ
P (x)= xP (x) – P (x).
ν+1 ν ν–1
ν – µ +1 ν – µ +1
Wronskians:
ν+µ+1 ν+µ+2
1 µ µ k 2µ Γ 2 Γ 2
W(P ν , Q ν )= , W(P , Q )= , k =2 ν–µ+1 ν–µ+2 .
ν
ν
1 – x 2 1 – x 2 Γ Γ
2 2
For n =0, 1, 2, ... ,
d n n n 2 n/2 d n
n
n
2 n/2
P (x)=(–1) (1 – x ) P ν (x), Q (x)=(–1) (1 – x ) Q ν (x).
ν n ν n
dx dx
Legendre polynomials
The Legendre polynomials P n (x) and the Legendre functions Q n (x) are defined by the formulas
n
1 d n 2 n 1 1+ x 1
P n (x)= (x – 1) , Q n (x)= P n (x)ln – P m–1 (x)P n–m (x).
n
n!2 dx n 2 1 – x m
m=1
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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