Example 21.1.2. The integral fo cos z dz/2², where C is the circle |2| = 1, is zero because 22n (-1)". (2n)! 1 1 + 4! COS Z z2 22 n=0 z2 = 0 (no 1/z term in the Laurent expansion). Therefore, by Equation (21.1) yields b1 the integral must vanish. When C is the circle |2| = 2, fce² dz/(z – 1)³ = ine because e = ee-1 Σ (z – 1)" n! (z – 1)² = e = e |1+ (z – 1) + 2! n=0 and 1 1 1 = e +.. (z – 1)2 (z – 1)3 Thus, b1 = e/2, and the integral is 2rib1 (z – 1)3 = ine.
Example 21.1.2. The integral fo cos z dz/2², where C is the circle |2| = 1, is zero because 22n (-1)". (2n)! 1 1 + 4! COS Z z2 22 n=0 z2 = 0 (no 1/z term in the Laurent expansion). Therefore, by Equation (21.1) yields b1 the integral must vanish. When C is the circle |2| = 2, fce² dz/(z – 1)³ = ine because e = ee-1 Σ (z – 1)" n! (z – 1)² = e = e |1+ (z – 1) + 2! n=0 and 1 1 1 = e +.. (z – 1)2 (z – 1)3 Thus, b1 = e/2, and the integral is 2rib1 (z – 1)3 = ine.
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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he question is about complex analysis and Calculus of Residues.
One of the 2 pictures is an example of a solution.
Please solve parts j, k and l.
Transcribed Image Text:Example 21.1.2. The integral f cos z dz/z², where C is the circle |z| = 1, is
zero because
22n
(-1)".
(2n)!
COS Z
1
1
22
-
z2
2
2
2
4!
n=0
0 (no 1/z term in the Laurent expansion). Therefore, by Equation (21.1)
yields bị
the integral must vanish.
When C is the circle |2|
2, fc e dz/(z – 1)³ = ine because
e = ee
(z – 1)"
= e |1+ (z – 1) +
2!
(z – 1)?
= e
n!
n=0
and
ez
1
1
1
= e
(z – 1)3
1)3
Thus, b1 = e/2, and the integral is 2ribi = ine.
(z – 1)2
2
-
Transcribed Image Text:21.1. Evaluate each of the following integrals, for all of which C is the circle
|2| = 3:
ez
dz.
(b) P z(2 – in)
COS Z
4z – 3
dz.
(c) P -
dz.
(a)
. 2(2 -
c z(z – 2)
22 +1
dz.
(d) P 2 – 1)
cosh z
dz.
+ 72
1
(f)
- COS Z
dz.
(e)
22
(:).
of
dz
dz.
(i) P 3(2+5)
sinh z
dz.
4
(g)
(h)
Z COS
dz.
dz
(1)
dz.
22
dz.
(j)
(k)
Jc sinh 2z
tan z dz.
dz
ez dz
(m)
dz.
22 sin z
(n)
(z – 1)(2 – 2)*
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