Evaluate the integral .2п 6²³ 0 I = cos(20) 5+4cos(0) - dᎾ
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
![Evaluate the integral
I
=
r2π cos (20)
-do
5 + 4cos (0)
(4)
Following the steps,
. Consider the polar representation of a complex
number z =
ree. What is the differential angle
element de in terms of dz? What is a natural con-
tour C that you want to integrate around? (Hint:
wouldn't it be nice if we could ignore the value of
r?)
• How can you write cos in terms of z? What about
cos(20)?
• Rewrite I in terms of a contour integral. Draw
your contours, and evaluate I using your preferred
method. Hints:
Multiply by = 1 to remove all negative pow-
ers of z in the numerator and denominator
- You will need to solve a quadratic equation,
but the solution should come out cleanly
=
Recall that for a function f(z) =
both p(z) and q(z) are analytic, Res[f(z
20)] = P(20)
q'
p(z) where
q(z)
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7be0278b-c62e-4a22-a818-58df3f094bc2%2F33520ee5-509d-4888-9bb2-99aec1ab4ff7%2Fpj7jq5x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Evaluate the integral
I
=
r2π cos (20)
-do
5 + 4cos (0)
(4)
Following the steps,
. Consider the polar representation of a complex
number z =
ree. What is the differential angle
element de in terms of dz? What is a natural con-
tour C that you want to integrate around? (Hint:
wouldn't it be nice if we could ignore the value of
r?)
• How can you write cos in terms of z? What about
cos(20)?
• Rewrite I in terms of a contour integral. Draw
your contours, and evaluate I using your preferred
method. Hints:
Multiply by = 1 to remove all negative pow-
ers of z in the numerator and denominator
- You will need to solve a quadratic equation,
but the solution should come out cleanly
=
Recall that for a function f(z) =
both p(z) and q(z) are analytic, Res[f(z
20)] = P(20)
q'
p(z) where
q(z)
=
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