5. Show that the integral in Example 6.7 simplifies to exp( 2 + | − 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:I
5. Show that the integral in Example 6.7 simplifies to exp[ 2 +
4
1.
![EXAMPLE 6.7 Let us give an exact calculation of the integral in Example
6.6. That is, we want fe exp z dz, where C is the line segment joining A = 0 to
. According to equation (2) of Section 1.6, we can parameterize C by
A
4
according to Theorem
B = 2 + i
2 + it, for (0) ≤ t ≤ 1. Since z'(t)
4),
6.1 we have that
z(t)
-
Lc exp z dz = f exp[(
= 2 +
So
i
= ( 2 + ²) ſ
- (2 + 4) L
||
π
2 + i
).] (₁
= 2 + i
π
2 +
e²tein/4 dt
dt
["e³[cos(π/4) + i sin(nt/4)] di
if e
e²¹cos(πt/4) dt + i
|
e21 sin(π/4) dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0faa0630-92f0-4653-b84e-0d22ebfc40eb%2F49a7412f-b895-4a05-b5bc-fb971af30895%2Fmg1lx3g_processed.png&w=3840&q=75)
Transcribed Image Text:EXAMPLE 6.7 Let us give an exact calculation of the integral in Example
6.6. That is, we want fe exp z dz, where C is the line segment joining A = 0 to
. According to equation (2) of Section 1.6, we can parameterize C by
A
4
according to Theorem
B = 2 + i
2 + it, for (0) ≤ t ≤ 1. Since z'(t)
4),
6.1 we have that
z(t)
-
Lc exp z dz = f exp[(
= 2 +
So
i
= ( 2 + ²) ſ
- (2 + 4) L
||
π
2 + i
).] (₁
= 2 + i
π
2 +
e²tein/4 dt
dt
["e³[cos(π/4) + i sin(nt/4)] di
if e
e²¹cos(πt/4) dt + i
|
e21 sin(π/4) dt
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