If nis odd, say n= 2m− 1 , then equation (12.94) eventually becomes
S 2 m− 1 =
(
2 m− 2
2 m− 1
)(
2 m− 4
2 m− 3
)
···
(
4
5
)(
2
3
)
·S 1 (12 .95)
where
S 1 =
∫π/ 2
0
sin x dx =−cos x]π/ 0 2 = 1
The numerator of equation (12.95) is a product of even integers and the denominator
of equation (12.95) is a product of odd integers so that one can employ the results
from equations (12.90) and (12.92) to write equation (12.95) in the form
S 2 m− 1 =
√
π
2
Γ(m)
Γ
( 2 m+1
2
) (12 .96)
If nis even, say n= 2m, then equation (12.94) eventually becomes
S 2 m=
(
2 m− 1
2 m
)(
2 m− 3
2 m− 2
)
···
(
5
6
)(
3
4
)(
1
2
)
S 0 (12 .97)
where
S 0 =
∫π/ 2
0
dx =π
2
(12 .98)
Note that the numerator in equation (12.97) is a product of odd integers and the
denominator is a product of even integers. Using the results from equations (12.90)
and (12.92) the above result can be expressed in the form
S 2 m=√^1
π
Γ
( 2 m+1
2
)
Γ(m+ 1)
π
2
(12 .99)
Example 12-6.
Let Cn =
∫π/ 2
0
cosnx dx and follow the step-by-step analysis as in the previous
example and demonstrate that
Cn=
∫π/ 2
0
cosnx dx =
√π
2
Γ(m)
Γ(^2 m 2 +1) if n= 2m−^1 is odd
√^1
π
Γ(^2 m 2 +1)
Γ(m+1)
π
2 if n= 2mis even
(12.100)
