Infinite Products
Γ
(
z
)
=
lim
k
→
∞
k
!
k
z
z
(
z
+
1
)
⋯
(
z
+
k
)
z
≠
0
,
-
1
,
-
2
,
…
,
1
Γ
(
z
)
=
z
ⅇ
γ
z
∏
k
=
1
∞
(
1
+
z
k
)
ⅇ
-
z
k
|
Γ
(
x
)
Γ
(
x
+
ⅈ
y
)
|
2
=
∏
k
=
0
∞
(
1
+
y
2
(
x
+
k
)
2
)
,
x
≠
0
,
-
1
,
…
.
∑
k
=
1
m
a
k
=
∑
k
=
1
m
b
k
then
∏
k
=
0
∞
(
a
1
+
k
)
(
a
2
+
k
)
⋯
(
a
m
+
k
)
(
b
1
+
k
)
(
b
2
+
k
)
⋯
(
b
m
+
k
)
=
Γ
(
b
1
)
Γ
(
b
2
)
⋯
Γ
(
b
m
)
Γ
(
a
1
)
Γ
(
a
2
)
⋯
Γ
(
a
m
)
provided that none of the
b
k
is zero or a negative integer.
