Page 783 - Handbook Of Integral Equations

Integral representations.
The functions I ν (x) and K ν (x) can be represented in terms of definite integrals:

x ν 1 2 ν–1/2 1
I ν (x)= 1 exp(–xt)(1 – t ) dt (x >0, ν > – ),
2
2 Γ(ν + )
π 1/2 ν –1
2
∞

K ν (x)= exp(–x cosh t) cosh(νt) dt (x > 0),
0
1 ∞
K ν (x)= 1 cos(x sinh t) cosh(νt) dt (x >0, –1< ν < 1),
cos πν 0
2
1 ∞
K ν (x)= 1 sin(x sinh t) sinh(νt) dt (x >0, –1< ν < 1).
sin πν 0
2
For integer ν = n,
π
1

I n (x)= exp(x cos t) cos(nt) dt (n =0, 1, 2, ... ),
π
0

∞ ∞ cos(xt)
K 0 (x)= cos(x sinh t) dt = √ dt (x > 0).
2
0 0 t +1
Integrals with modified Bessel functions
x x λ+ν+1 λ + ν +1 λ + ν +3 x 2
λ
x I ν (x) dx = ν F , , ν +1; , Re(λ+ν)> –1,
0 2 (λ + ν +1)Γ(ν +1) 2 2 4
where F(a, b, c; x) is the hypergeometric series (see Section 10.9 of this supplement),
x 2 ν–1 Γ(ν) λ – ν +1 λ – ν +3 x 2
λ
x K ν (x) dx = x λ–ν+1 F ,1 – ν, ,
0 λ – ν +1 2 2 4
2 –ν–1 Γ(–ν) λ+ν+1 λ + ν +1 λ + ν +3 x 2
+ x F ,1 + ν, , , Re λ > |Re ν| – 1.
λ + ν +1 2 2 4

Asymptotic expansions as x →∞
M
2
2
2
2
2
e x m (4ν – 1)(4ν – 3 ) ... [4ν – (2m – 1) ]
I ν (x)= √ 1+ (–1) m ,
2πx m!(8x)
m=1
M
2 2 2 2 2
π –x (4ν – 1)(4ν – 3 ) ... [4ν – (2m – 1) ]
K ν (x)= e 1+ .
2x m!(8x) m
m=1
The terms of the order of O(x –M–1 ) are omitted in the braces.
10.8. Degenerate Hypergeometric Functions
Definitions. Basic Formulas
The degenerate hypergeometric functions Φ(a, b; x) and Ψ(a, b; x) are solutions of the degenerate
hypergeometric equation

xy xx +(b – x)y – ay =0.
x

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 768