Page 788 - Handbook Of Integral Equations

Basic properties
The function F possesses the following properties:

F(α, β, γ; x)= F(β, α, γ; x),
F(α, β, γ; x)=(1 – x) γ–α–β F(γ – α, γ – β, γ; x),
x

–α
F(α, β, γ; x)=(1 – x) F α, γ – β, γ; ,
x – 1
d n (α) n (β) n
F(α, β, γ; x)= F(α + n, β + n, γ + n; x).
dx n (γ) n
If γ is not an integer, then the general solution of the hypergeometric equation can be written in
the form
y = C 1 F(α, β, γ; x)+ C 2 x 1–γ F(α – γ +1, β – γ +1, 2 – γ; x).

Integral representations
For γ > β > 0, the hypergeometric function can be expressed in terms of a definite integral:
Γ(γ) 1 β–1 γ–β–1 –α
F(α, β, γ; x)= t (1 – t) (1 – tx) dt,
Γ(β) Γ(γ – β) 0
where Γ(β) is the gamma function.
See M. Abramowitz and I. Stegun (1979) and H. Bateman and A. Erd´ elyi (1973, Vol. 1) for
more detailed information about hypergeometric functions.

10.10. Legendre Functions

Definitions. Basic formulas
µ
µ
The associated Legendre functions P (z) and Q (z)ofthe first and the second kind are linearly
ν ν
independent solutions of the Legendre equation:
2 –1
2
2
(1 – z )y – 2zy +[ν(ν +1) – µ (1 – z ) ]y =0,

zz z
where the parameters ν and µ and the variable z can assume arbitrary real or complex values.
For |1 – z| < 2, the formulas
1 z +1 µ/2 1 – z
µ
P (z)= F –ν,1 + ν,1 – µ, ,
ν
Γ(1 – µ) z – 1 2
µ µ
z – 1 2 1 – z z +1 2 1 – z

µ
Q (z)= A F –ν,1 + ν,1 + µ, + B F –ν,1 + ν,1 – µ, ,
ν
z +1 2 z – 1 2
Γ(–µ) Γ(1 + ν + µ) iµπ Γ(µ) 2
iµπ
A = e , B = e , i = –1,
2 Γ(1 + ν – µ) 2
are valid, where F(a, b, c; z) is the hypergeometric series (see (see Section 10.9 of this supplement).
For |z| >1,
1
2 –ν–1 Γ(– – ν) 1+ ν – µ 2+ ν – µ 2ν +3 1
µ 2 –ν+µ–1 2 –µ/2
P (z)= √ z (z – 1) F , , ,
ν 2
π Γ(–ν – µ) 2 2 2 z
1
ν
2 Γ( + ν) ν+µ ν + µ 1 – ν – µ 1 – 2ν 1
2
+ √ 2 z (z – 1) –µ/2 F – , , , ,
π Γ(1 + ν – µ) 2 2 2 z 2
√
π Γ(ν + µ +1) –ν–µ–1 2 µ/2 2+ ν + µ 1+ ν + µ 2ν +3 1
iπµ
µ
Q (z)= e 3 z (z – 1) F , , , .
ν
2 ν+1 Γ(ν + ) 2 2 2 z 2
2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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