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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 28 #28
12-28 Real Analysis
and ψ(0) does not exist. As before,
1
0
ψ(y)dy =−
1
2
. Hence
1
0
1
0
x y
(x + y)
3
dx
dy =−
1
2
.
This shows that iterated integrals exist but are not equal. Since the
function f (x, y) is not bounded in the neighbourhood of the origin (0, 0),
the double integral does not exist.
The following example shows that for a function f of two variables

S
f (x, y) dx dy need not exist but both the iterated integrals exist and
can be equal.
Example 12.7.7 Let S =[0, 1]×[0, 1] and E =
k=1
E(p
k
), where
E(p
k
) =

n
p
k
,
m
p
k
/n = 1, 2, 3, ..., p
k
1, m = 1, 2, 3, ..., p
k
1
for each k
th
prime number p
k
.
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Publisher Resources

ISBN: 9781299447561Publisher Website