1.6 Applications: Examples
13
1.6
Applications: Examples
The Γ - and incomplete gamma functions serve quite frequently as normalization or
scaling factors. Due to their relation to different integrals they are also useful for
computing integrals. For example, Γ (z) holds the following relation:
∞
0
x
α exp(−βx
2 )dx =
Γ (
α+1
2 )
2β (α+1)/2 .
(1.28)
The exponential integrals can be easily calculated with the help of the incomplete
gamma function Γ (a, x).
E n (x) =
∞
1
exp(−x t)
t n
x > 0 n ∈ N,
(1.29a)
E n (x) = x
n−1 Γ (x, 1 − n),
(1.29b)
thus
E 0 (x) =
exp(−x)
x
and E n (0) =
1
n − 1
n > 1.
(1.29c)
As examples we will have a closer look on the Böhmer integrals.
1.6.1 Examples
The Böhmer Integrals
The Böhmer or generalized sin and cos Integrals are given by
si(z, a) =
∞
z
t
a−1 sin(t)dt (a) < 1
(1.30)
ci(z, a) =
∞
z
t
a−1 cos(t)dt (a) < 1
(1.31)
Si(z, a) =
z
0
t
a−1 sin(t)dt (a) > −1
(1.32)
Ci(z, a) =
z
0
t
a−1 cos(t)dt (a) > 0,
(1.33)
and hold the following relations:
ci(z, a) ± i si(z, a) = exp
±
1
2
iπa
Γ (∓i z, a)
(1.34)