This problem extends Example 5.14 and Exercise 5.19, allowing us to compute the following integrals simultaneously: For m e N, consider sin(x) dr x2m + 4280 cos(r) dr. r2m + 4280 and (a) Let f(2) = R> V4280. and z2m + 4280 Compute ff where or is the positively oriented semicircle formed by the line segment [-R, R] on the real axis, followed by the circular arc of radius R in the upper half plane from R to –R. DRAW STUFF, but don't prove anything for part (a). (b) Prove that limr-∞ Se. ƒ = 0. HINTS: Show |f(2)| < 2 for sufficiently large |z|. How large is |ei=| in the upper half plane?
This problem extends Example 5.14 and Exercise 5.19, allowing us to compute the following integrals simultaneously: For m e N, consider sin(x) dr x2m + 4280 cos(r) dr. r2m + 4280 and (a) Let f(2) = R> V4280. and z2m + 4280 Compute ff where or is the positively oriented semicircle formed by the line segment [-R, R] on the real axis, followed by the circular arc of radius R in the upper half plane from R to –R. DRAW STUFF, but don't prove anything for part (a). (b) Prove that limr-∞ Se. ƒ = 0. HINTS: Show |f(2)| < 2 for sufficiently large |z|. How large is |ei=| in the upper half plane?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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part B
![This problem extends Example 5.14 and Exercise 5.19, allowing us to compute the following integrals
simultaneously: For m e N, consider
sin(x)
dr
x2m + 4280
cos(r)
dr.
r2m + 4280
and
(a) Let f(2) =
R> V4280.
and
z2m + 4280
Compute ff where og is the positively oriented semicircle formed by the line segment [-R, R]
on the real axis, followed by the circular arc of radius R in the upper half plane from R to –R.
DRAW STUFF, but don't prove anything for part (a).
(b) Prove that limr-∞ Sa. f = 0. HINTS: Show |f(2)| <
2
for sufficiently large |2|.
|z/2m
How large is |ei=| in the upper half plane?
(c)Use the results of (a) and (b) to compute the integrals below and prove your
computations are correct.
sin(x)
dr
a2m + 4280
cos(r)
and
dr.
a2m + 4280
(One of these can be computed quickly using an idea from calculus, by the way.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5958936e-64d8-4f9f-ac16-af104f1eb32d%2F275265f3-ef0d-48ee-982b-d934bc36e77f%2Fewwmltw_processed.png&w=3840&q=75)
Transcribed Image Text:This problem extends Example 5.14 and Exercise 5.19, allowing us to compute the following integrals
simultaneously: For m e N, consider
sin(x)
dr
x2m + 4280
cos(r)
dr.
r2m + 4280
and
(a) Let f(2) =
R> V4280.
and
z2m + 4280
Compute ff where og is the positively oriented semicircle formed by the line segment [-R, R]
on the real axis, followed by the circular arc of radius R in the upper half plane from R to –R.
DRAW STUFF, but don't prove anything for part (a).
(b) Prove that limr-∞ Sa. f = 0. HINTS: Show |f(2)| <
2
for sufficiently large |2|.
|z/2m
How large is |ei=| in the upper half plane?
(c)Use the results of (a) and (b) to compute the integrals below and prove your
computations are correct.
sin(x)
dr
a2m + 4280
cos(r)
and
dr.
a2m + 4280
(One of these can be computed quickly using an idea from calculus, by the way.)
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