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or
q
−
1
B
(
p
,
q
)=
1
B
(
p
,
q
−
1
)
.
(A.35)
p
+
q
−
When
q
>
0and
p
>
1, by the symmetry of the
B
-function we have
p
−
1
B
(
p
,
q
)=
1
B
(
p
−
1
,
q
)
.
(A.36)
+
−
p
q
Equations (A.35) and (A.36) lead to
(
p
−
1
)(
q
−
1
)
B
(
p
,
q
)=
B
(
p
−
1
,
q
−
1
)
.
(A.37)
(
p
+
q
−
1
)(
p
+
q
−
2
)
cos
2
=
Let
x
θ
; we obtain another form of the
B
-function
2
2
cos
2
p
−
1
sin
2
q
−
1
B
(
p
,
q
)=
θ
θ
d
θ
.
0
Thus
B
1
1
2
1
2
,
=
π
.Let
x
=
t
. Thus
1
+
1
+
∞
t
q
−
1
t
q
−
1
B
(
p
,
q
)=
p
+
q
d
t
+
p
+
q
d
t
,
(
1
+
t
)
(
1
+
t
)
0
1
which reduces into, by a variable transformation
t
=
1
/
u
in the second integral,
1
t
p
−
1
t
q
−
1
+
B
(
p
,
q
)=
q
d
t
.
(
1
+
t
)
p
+
0
The Euler integral of the second kind is a generalized integral that contains positive
parameter
x
,
+
∞
e
−
t
t
x
−
1
d
t
Γ
(
x
)=
,
x
>
0
.
(A.38)
0
It is called the
Gamma function
,the
Γ
-function
for short, and satisfies
Γ
(
x
+
1
)=
x
Γ
(
x
)
.
When
x
≤
0, the integral in Eq. (A.38) is divergent. We thus define
Γ
(
x
)=
Γ
(
x
+
1
)
/
x
, −
1
<
x
<
0
.
(A.39)
This definition is also valid for
−
2
<
x
< −
1etc.However,
Γ
(
x
)
→
∞
as
x
tends to
0,
−
1,
−
2,
···
. Therefore, the
Γ
-function is defined by
⎧
⎨
+
∞
e
−
t
t
x
−
1
d
t
,
x
>
0
,
Γ
(
x
)=
0
)
x
⎩
Γ
(
x
+
1
,
x
<
0
,
x
=
−
1
,−
2
, ···.
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