(a) Use Cauchy's Residue Theorem to evaluate log-/22 dz. (2²+1)² Fr.R (b) Show in detail exactly one of the following. dz⇒0 dz • √ (2² + 1)² 6 log-/22 dz0 as R→∞ • Sm (2² + 1)² 0 log-/22 dz0 as r 0+ d
(a) Use Cauchy's Residue Theorem to evaluate log-/22 dz. (2²+1)² Fr.R (b) Show in detail exactly one of the following. dz⇒0 dz • √ (2² + 1)² 6 log-/22 dz0 as R→∞ • Sm (2² + 1)² 0 log-/22 dz0 as r 0+ d
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
answer a and b
![Define the branch of the logarithm log-/2 given by
Recall that log-/2 is defined for all nonzero z € C, but it is only analytic on
=C\ {-iy: y ≥ 0), with derivative for z EN.
z
Let 0 <r < 1< R. Define Yr(t) = Ret for t = [0, ]. Let TR be the closed curve
given by [r, R] UYRU[-R, -r] U-% (where - is the same curve % but traced
in the opposite direction).
(a) Use Cauchy's Residue Theorem to evaluate
log-/22
Sar
(b) Show in detail exactly one of the following.
log-T/2z
(2² + 1)²
dz0 as R→∞
dz0 as r0+
●
log-/22= In |2|+i arg-/22, arg-/22 E
(c) Write
log-/22
(22 + 1)²
•La
DO In x
log-/22
(√ng + √(1-1-1) ) ( 2² + 1)² d
dz
as a single integral with respect to the positive real variable z.
(d) From the previous results, deduce that
5
z € (-1,77).
dz.
π
dx = - and
2+1)²
(22
50%
1
(x² + 1)²
dx =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa221a80a-8c40-4680-bd4f-37dfd9fe3c2c%2Feee5cb00-c544-4ce6-8b2f-ec881cd29a62%2Fa9u3yjr_processed.png&w=3840&q=75)
Transcribed Image Text:Define the branch of the logarithm log-/2 given by
Recall that log-/2 is defined for all nonzero z € C, but it is only analytic on
=C\ {-iy: y ≥ 0), with derivative for z EN.
z
Let 0 <r < 1< R. Define Yr(t) = Ret for t = [0, ]. Let TR be the closed curve
given by [r, R] UYRU[-R, -r] U-% (where - is the same curve % but traced
in the opposite direction).
(a) Use Cauchy's Residue Theorem to evaluate
log-/22
Sar
(b) Show in detail exactly one of the following.
log-T/2z
(2² + 1)²
dz0 as R→∞
dz0 as r0+
●
log-/22= In |2|+i arg-/22, arg-/22 E
(c) Write
log-/22
(22 + 1)²
•La
DO In x
log-/22
(√ng + √(1-1-1) ) ( 2² + 1)² d
dz
as a single integral with respect to the positive real variable z.
(d) From the previous results, deduce that
5
z € (-1,77).
dz.
π
dx = - and
2+1)²
(22
50%
1
(x² + 1)²
dx =
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