Page 786 - Handbook Of Integral Equations

Integral representations

1

Γ(b) xt a–1 b–a–1
Φ(a, b; x)= e t (1 – t) dt (for b > a > 0),
Γ(a) Γ(b – a) 0
1 ∞ –xt a–1 b–a–1
Ψ(a, b; x)= e t (1 + t) dt (for a >0, x > 0),
Γ(a)
0
where Γ(a) is the gamma function.
Integrals with degenerate hypergeometric functions

b – 1

Φ(a, b; x) dx = Ψ(a – 1, b – 1; x)+ C,
a – 1
1

Ψ(a, b; x) dx = Ψ(a – 1, b – 1; x)+ C,
1 – a
n+1
k+1 n–k+1
(–1) (1 – b) k x
n
x Φ(a, b; x) dx = n! Φ(a – k, b – k; x)+ C,
(1 – a) k (n – k + 1)!
k=1
n+1
k+1 n–k+1
(–1) x
n
x Ψ(a, b; x) dx = n! Ψ(a – k, b – k; x)+ C.
(1 – a) k (n – k + 1)!
k=1
Asymptotic expansion as |x| →∞.
N
Γ(b) x a–b (b – a) n (1 – a) n –n
Φ(a, b; x)= e x x + ε , x >0,
Γ(a) n!
n=0
N
Γ(b) –a (a) n (a – b +1) n –n
Φ(a, b; x)= (–x) (–x) + ε , x <0,
Γ(b – a) n!
n=0
N
Ψ(a, b; x)= x –a (–1) n (a) n (a – b +1) n x –n + ε , –∞ < x < ∞,
n!
n=0
where ε = O(x –N–1 ).
10.9. Hypergeometric Functions
Definition
The hypergeometric functions F(α, β, γ; x) is a solution the Gaussian hypergeometric equation

x(x – 1)y +[(α + β +1)x – γ]y + αβy =0.

xx
x
For γ ≠ 0, –1, –2, –3, ... , the function F(α, β, γ; x) can be expressed in terms of the hypergeo-
metric series:
∞ k
(α) k (β) k x
F(α, β, γ; x)=1 + , (α) k = α(α +1) ... (α + k – 1),
(γ) k k!
k=1
which certainly converges for |x| <1.
Table S2 shows some special cases when F can be expressed in term of elementary functions.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 771