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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_10” 2011/5/22 23:26 page 22 #22
10-22 Real Analysis
Corollary 10.4.6
β(x, y) =
(x)(y)
(x + y)
= 2
π/2
0
(sin θ)
2x1
(cos θ)
2y1
dθ
and
1
2
=
π.
Proof Substitute t = sin
2
θ in β(x, y) =
1
0
t
x1
(1 t)
y1
dt and get
the required equality. Further the special case x = y =
1
2
gives
β
1
2
,
1
2
=
1
2

2
= 2
π/2
0
dθ = π
which in turn gives,
1
2
=
π.
Theorem 10.4.7 (x) = 2
0
s
2x1
e
s
2
ds (0 < s < ) and
−∞
e
s
2
ds =
π.
Proof In the definition of (x), put t = s
2
to get the required
result. The choice x =
1
2
gives,
−∞
e
s
2
ds = 2
0
e
s
2
ds =
1
2
=
π.
Theorem 10.4.8
(x) =
2
x1
π
x
2
x + 1
2
.
Proof We only need to verify that
f (x) =
2
x1
π
x
2
x +
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Publisher Resources

ISBN: 9781299447561Publisher Website