Page 778 - Handbook Of Integral Equations

Recurrent formulas
α –x
γ(α +1, x)= αγ(α, x) – x e ,
α –x
Γ(α +1, x)= αΓ(α, x)+ x e .

Asymptotic expansions as x → 0:
∞ n α+n
(–1) x
γ(α, x)= ,
n!(α + n)
n=0
∞ n α+n
(–1) x
Γ(α, x)= Γ(α) – .
n!(α + n)
n=0
Asymptotic expansions as x →∞:

M–1
(1 – α) m
e
γ(α, x)= Γ(α) – x α–1 –x + O |x| –M ,
(–x) m
m=0
M–1
(1 – α) m –M 3 3
α–1 –x
Γ(α, x)= x e + O |x| – π <arg x < 2 .
2
(–x) m
m=0
Integral functions related to the gamma function:
1 1 2 1 1 2
erf x = √ γ , x , erfc x = √ Γ , x , Ei(–x)= –Γ(0, x).
π 2 π 2
Incomplete beta function:
1

B x (p, q)= t p–1 (1 – t) q–1 dt,
0
where Re x > 0 and Re y >0.

10.6. Bessel Functions

Definition and basic formulas
The Bessel function of the first kind, J ν (x), and the Bessel function of the second kind, Y ν (x)
(also called the Neumann function), are solutions of the Bessel equation
2
2

2
x y + xy +(x – ν )y =0
xx x
and are defined by the formulas
∞ k ν+2k
(–1) (x/2) J ν (x) cos πν – J –ν (x)
J ν (x)= , Y ν (x)= . (1)
k! Γ(ν + k +1) sin πν
k=0
The formula for Y ν (x) is valid for ν ≠ 0, ±1, ±2, ... (the cases ν ≠ 0, ±1, ±2, ... are discussed in
what follows).
The general solution of the Bessel equation has the form Z ν (x)= C 1 J ν (x)+ C 2 Y ν (x) and is
called the cylinder function.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 763