Page 773 - Handbook Of Integral Equations

Pochhammer symbol (k =1, 2, ... )

Γ(a + n) n Γ(1 – a)
(a) n = a(a +1) ... (a + n – 1) = =(–1) ,
Γ(a) Γ(1 – a – n)
(n + k – 1)!
(a) 0 =1, (a) n+k =(a) n (a + n) k , (n) k = ,
(n – 1)!
Γ(a – n) (–1) n
(a) –n = = , where a ≠ 1, ... , n;
Γ(a) (1 – a) n
(2n)! –2n (2n + 1)!
–2n
(1) n = n!, (1/2) n =2 , (3/2) n =2 ,
n! n!
(a) mk+nk (a) 2n (a) k (a + k) n
(a + mk) nk = , (a + n) n = , (a + n) k = .
(a) mk (a) n (a) n
Bernoulli numbers, B n
Definition:
∞
x x n
= B n .
x
e – 1 n!
n=0
The numbers:
1
1
B 0 =1, B 1 = – , B 2 = , B 4 = – 1 , B 6 = 1 , B 8 = – 1 , B 10 = 5 , ... ,
2 6 30 42 30 66
B 2m+1 = 0 for m =1, 2, ...

10.2. Error Functions and Integral Exponent

Error function and complementary error function (probability integrals)
Definitions:
x ∞

2 2 2 2
erf x = √ exp(–t ) dt, erfc x =1 – erf x = √ exp(–t ) dt.
π π
0 x
Expansion of erf x into series in powers of x as x → 0:
∞ 2k+1 ∞ k 2k+1
2 k x 2 2 2 x
erf x = √ (–1) = √ exp –x .
π (k)!(2k +1) π 2k + 1)!!
k=0 k=0
Asymptotic expansion of erfc x as x →∞:

M–1
1
1 2 m 2 m –2M–1
erfc x = √ exp –x (–1) + O |x| , M =1, 2, ...
π x 2m+1
m=0
Integral exponent
Definition:
x
t
e
Ei(x)= dt for x <0,
t
–∞
–ε e t x e t

Ei(x) = lim dt + dt for x >0.
ε→+0 t ε t
–∞
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 758