Digamma function
logarithmic derivative of the gamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
ψ
(
z
)
=
d
d
z
ln
Γ
(
z
)
=
Γ
′
(
z
)
Γ
(
z
)
.
{\displaystyle \psi (z)={\frac {d}{dz}}\ln \Gamma (z)={\frac {\Gamma '(z)}{\Gamma (z)}}.}
It is the first of the polygamma functions. This function is strictly increasing and strictly concave on
(
0
,
∞
)
{\displaystyle (0,\infty )}
, and it asymptotically behaves as
ψ
(
z
)
∼
ln
z
−
1
2
z
,
{\displaystyle \psi (z)\sim \ln {z}-{\frac {1}{2z}},}
for complex numbers with large modulus (
|
z
|
→
∞
{\displaystyle |z|\rightarrow \infty }
) in the sector
|
arg
z
|
<
π
−
ε
{\displaystyle \left|\arg z\right|<\pi -\varepsilon }
for any
ε
>
0
{\displaystyle \varepsilon >0}
.
The digamma function is often denoted as
ψ
0
(
x
)
,
ψ
(
0
)
(
x
)
{\displaystyle \psi _{0}(x),\psi ^{(0)}(x)}
or Ϝ (the uppercase form of the archaic Greek letter digamma meaning double-gamma).
Relation to harmonic numbers
The gamma function obeys the equation
Γ
(
z
+
1
)
=
z
Γ
(
z
)
.
{\displaystyle \Gamma (z+1)=z\Gamma (z).\,}
Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives:
log
Γ
(
z
+
1
)
=
log
(
z
)
+
log
Γ
(
z
)
,
{\displaystyle \log \Gamma (z+1)=\log(z)+\log \Gamma (z),}
Differentiating both sides with respect to z gives:
ψ
(
z
+
1
)
=
ψ
(
z
)
+
1
z
{\displaystyle \psi (z+1)=\psi (z)+{\frac {1}{z}}}
Since the harmonic numbers are defined for positive integers n as
H
n
=
∑
k
=
1
n
1
k
,
{\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k}},}
the digamma function is related to them by
ψ
(
n
)
=
H
n
−
1
−
γ
,
{\displaystyle \psi (n)=H_{n-1}-\gamma ,}
where H0 = 0, and γ is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values
ψ
(
n
+
1
2
)
=
−
γ
−
2
ln
2
+
∑
k
=
1
n
2
2
k
−
1
=
−
γ
−
2
ln
2
+
2
H
2
n
−
H
n
.
{\displaystyle \psi \left(n+{\tfrac {1}{2}}\right)=-\gamma -2\ln 2+\sum _{k=1}^{n}{\frac {2}{2k-1}}=-\gamma -2\ln 2+2H_{2n}-H_{n}.}
Integral representations
If the real part of z is positive then the digamma function has the following integral representation due to Gauss:
ψ
(
z
)
=
∫
0
∞
(
e
−
t
t
−
e
−
z
t
1
−
e
−
t
)
d
t
.
{\displaystyle \psi (z)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-zt}}{1-e^{-t}}}\right)\,dt.}
Combining this expression with an integral identity for the Euler–Mascheroni constant
γ
{\displaystyle \gamma }
gives:
ψ
(
z
+
1
)
=
−
γ
+
∫
0
1
(
1
−
t
z
1
−
t
)
d
t
.
{\displaystyle \psi (z+1)=-\gamma +\int _{0}^{1}\left({\frac {1-t^{z}}{1-t}}\right)\,dt.}
The integral is Euler's harmonic number
H
z
{\displaystyle H_{z}}
, so the previous formula may also be written
ψ
(
z
+
1
)
=
ψ
(
1
)
+
H
z
.
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