Definitions
Contents
- Gamma and Psi Functions
- Euler's Constant
- Pochhammer's Symbol
Gamma and Psi Functions
Euler's Integral
Γ
(
z
)
=
∫
0
∞
ⅇ
-
t
t
z
-
1
ⅆ
t
ℜ
z
>
0
.
When
ℜ
z
≤
0
,
Γ
(
z
)
is defined by analytic continuation. It is a meromorphic
function with no zeros, and with simple poles of residue
(
-
1
)
n
n
!
at
z
=
-
n
.
1
Γ
(
z
)
is entire, with simple zeros at
z
=
-
n
.
ψ
(
z
)
=
Γ
′
(
z
)
Γ
(
z
)
z
≠
0
,
-
1
,
-
2
,
…
ψ
(
z
)
is meromorphic with simple poles of residue
-
1
at
z
=
-
n
.
Euler's Constant
γ
=
lim
n
→
∞
(
1
+
1
2
+
1
3
+
…
+
1
n
-
ln
n
)
=
0.57721 56649 01532 86060
…
Pochhammer's Symbol
(
a
)
0
=
1
(
a
)
n
=
a
(
a
+
1
)
(
a
+
2
)
⋯
(
a
+
n
-
1
)
(
a
)
n
=
Γ
(
a
+
n
)
Γ
(
a
)
a
≠
-
n
,
-
n
-
1
,
-
n
-
2
,
…
