Page 777 - Handbook Of Integral Equations

Logarithmic derivative of the gamma function
Definition:

ln Γ(z) Γ (z)
z
ψ(z)= = .
dz Γ(z)
Functional relations:
1
ψ(z) – ψ(1 + z)= – ,
z
ψ(z) – ψ(1 – z)= –π cot(πz),
1
ψ(z) – ψ(–z)= –π cot(πz) – ,
z
1 1
ψ + z – ψ – z = π tan(πz),
2 2
1 m–1 k

ψ(mz)=ln m + ψ z + .
m m
k=0
Integral representations (Re z > 0):
∞

–t –z –1
ψ(z)= e – (1 + t) t dt,
0
∞

–1 –t –1 –tz
ψ(z)=ln z + t – (1 – e ) e dt,
0
1
1 – t z–1
ψ(z)= –C + dt,
1 – t
0
where C = –ψ(1) = 0.5572 ... is the Euler constant.
Values for integer argument:
n–1
–1
ψ(1) = –C, ψ(n)= –C + k (n =2, 3, ... )
k=1
Beta function
Definition:
1
B(x, y)= t x–1 (1 – t) y–1 dt,
0
where Re x > 0 and Re y >0.
Relationship with the gamma function:

Γ(x)Γ(y)
B(x, y)= .
Γ(x + y)

10.5. Incomplete Gamma Function

Definitions. Integral representations

x

–t α–1
γ(α, x)= e t dt, Re α >0,
0
∞

–t α–1
Γ(α, x)= e t dt = Γ(α) – γ(α, x).
x
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 762