4
1 Mathematical Physics
Γ(n) =
∞
0
e
−x x
n−1 dx (Re n > 0)
(1.11)
Γ(n + 1) = nΓ(n)
(1.12)
If n is a positive integer
Γ(n + 1) = n!
(1.13)
Γ
1
2
=
√
π; Γ
3
2
=
√
π
2
; Γ
5
2
=
3
4
√
π
(1.14)
Γ
n +
1
2
=
1.3.5 . . . (2n − 1)
√
π
2 n
(n = 1, 2, 3, . . .)
(1.15)
Γ
−n +
1
2
=
(−1)
n 2
n
√
π
1.3.5 . . . (2n − 1)
(n = 1, 2, 3, . . .)
(1.16)
Γ(n + 1) = n! ∼ =
√
2πn n
n e
−n
(Stirling’s formula)
(1.17)
n → ∞
Beta function B(m, n) is defined as
B(m, n) =
Γ(m)Γ(n)
Γ(m + n)
(1.18)
B(m, n) = B(n, m)
(1.19)
B(m, n) = 2
π/2
0
sin
2m−1
θ cos
2n−1
θ dθ
(1.20)
B(m, n) =
∞
0
t
m−1
(1 + t) m+n dt
(1.21)
Special funtions, properties and differential equations
Hermite functions:
Differential equation:
y
− 2x y
+ 2ny = 0
(1.22)
when n = 0, 1, 2, . . . then we get Hermite’s polynomials H n (x) of degree n, given
by
H n (x) = (−1)
n e
x
2 d
n
dx n
e
−x
2
(Rodrigue’s formula)