Functional Relations
Contents
- Recurrence
- Reflection
- Multiplication
- Bohr-Mollerup Theorem
Recurrence
Γ
(
z
+
1
)
=
z
Γ
(
z
)
ψ
(
z
+
1
)
=
ψ
(
z
)
+
1
z
Reflection
Γ
(
z
)
Γ
(
1
-
z
)
=
π
sin
(
π
z
)
z
≠
0
,
ŷ
1
,
…
,
ψ
(
z
)
-
ψ
(
1
-
z
)
=
-
π
tan
(
π
z
)
z
≠
0
,
ŷ
1
,
…
.
Multiplication
2
z
≠
0
,
-
1
,
-
2
,
…
,
Γ
(
2
z
)
=
π
-
1
2
2
2
z
-
1
Γ
(
z
)
Γ
(
z
+
1
2
)
3
z
≠
0
,
-
1
,
-
2
,
…
,
Γ
(
3
z
)
=
(
2
π
)
-
1
3
3
z
-
(
1
2
)
Γ
(
z
)
Γ
(
z
+
1
3
)
Γ
(
z
+
2
3
)
n
z
≠
0
,
-
1
,
-
2
,
…
,
Γ
(
n
z
)
=
(
2
π
)
(
1
-
n
)
2
n
n
z
-
(
1
2
)
∏
k
=
0
n
-
1
Γ
(
z
+
k
n
)
∏
k
=
1
n
-
1
Γ
(
k
n
)
=
(
2
π
)
(
n
-
1
)
2
n
-
1
2
ψ
(
2
z
)
=
1
2
(
ψ
(
z
)
+
ψ
(
z
+
1
2
)
)
+
ln
2
ψ
(
n
z
)
=
1
n
∑
k
=
0
n
-
1
ψ
(
z
+
k
n
)
+
ln
n
Bohr-Mollerup Theorem
If a positive function
f
(
x
)
on
(
0
,
∞
)
satisfies
f
(
x
+
1
)
=
x
f
(
x
)
,
f
(
1
)
=
1
, and
ln
f
(
x
)
is convex, then
f
(
x
)
=
Γ
(
x
)
.