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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_12” 2011/5/23 1:04 page 23 #23
Real-valued Functions of Two Real Variables 12-23
Theorem 12.6.3 enables us to compute integrals, which involve a
parameter. The following example illustrates this.
Example 12.6.4 If |a| < 1, compute
π
0
log(1 + a cos x)
cos x
dx.
Let
f (a, x) =
log(1 + a cos x)
cos x
.
f (a, x) is discontinuous at x =
π
2
and
lim
x
π
2
log(1 + a cos x)
cos x
= a, f
a
(a, x) =
1
1 + a cos x
.
It is easy to show that f and f
a
are both continuous functions of
two variables in the domain {(a, x)/ 0 x π , |a| < 1}. Define
f (a, π/2) = a. Put
φ(a) =
π
0
f (a, x)dx. (12.8)
Then
φ
(a) =
π
0
f
a
(a, x)dx =
π
0
1
1 + a cos x
dx.
Putting tan(x/2) = t, we
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Publisher Resources

ISBN: 9781299447561Publisher Website