We study the residue of functions of the form g(z) = 8 in a domain D. The singularities of g(2) consists of the singularities of f as well as the zeros of f. We assume that f has no singularity other than poles. (a) of order m at z0, show that Suppose that f(z) is analytic in |z – zol < R, and has a zero Res 20 = m. (b) a pole of order n at zo, show that Suppose that f(2) is analytic in 0< z- zol < R, and has Res -п. (c) in a domain D, which does not pass through any pole or zero of f and encloses its interior region Nc D. Suppose that in 2 there are: • a pole z1 of order 3, a pole z2 of order 8, a pole z3 of order 1 of the function f; • a zero wi of order 2, a zero wz of order 4 of the function f. Suppose that is a positively oriented simple closed curve 1 Compute 2ni f'(2) -dz. f(2) Hint You might want to use the residue theorem and results from previous sub-questions to solve this sub-question. 2.
We study the residue of functions of the form g(z) = 8 in a domain D. The singularities of g(2) consists of the singularities of f as well as the zeros of f. We assume that f has no singularity other than poles. (a) of order m at z0, show that Suppose that f(z) is analytic in |z – zol < R, and has a zero Res 20 = m. (b) a pole of order n at zo, show that Suppose that f(2) is analytic in 0< z- zol < R, and has Res -п. (c) in a domain D, which does not pass through any pole or zero of f and encloses its interior region Nc D. Suppose that in 2 there are: • a pole z1 of order 3, a pole z2 of order 8, a pole z3 of order 1 of the function f; • a zero wi of order 2, a zero wz of order 4 of the function f. Suppose that is a positively oriented simple closed curve 1 Compute 2ni f'(2) -dz. f(2) Hint You might want to use the residue theorem and results from previous sub-questions to solve this sub-question. 2.
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
[Complex Variables] How do you solve this? Thanks
Hint: To solve (c), combine the result from (a) and (b) togehter
The second picture is the residue theorem
Transcribed Image Text:THEOREM 1 The Residue Theorem Suppose that f is analytic on a simply-connected domain
D except for a finite number of isolated singularities at points z1, ..., ZN of D. Let y
be a piecewise smooth positively oriented simple closed curve in D that does not
pass through any of the points z1,..., ZN. Then
|fe) d -2πί Σ Res (f; 7, .
Zk inside y
where the sum is taken over all those singularities z, of f that lie inside y.
Proof Let y1, ..., YN be disjoint positively oriented circles centered at z1, .., ZN,
respectively, chosen to be so small that the discs they bound are disjoint. We can
apply Green's Theorem to the function f on the domain whose boundary is y and
Y1,..., YN. Since f is analytic on 2, the value of the line integral is zero. Therefore,
f(2) dz – E| f(2) dz.
Yk
However, f, f(z) dz = 2ni Res(f; zk). This finishes the proof.
Transcribed Image Text:We study the residue of functions of the form g(z) = 8 in a
domain D. The singularities of g(2) consists of the singularities of f as
well as the zeros of f. We assume that f has no singularity other than
poles.
(a)
of order m at z0, show that
Suppose that f(z) is analytic in |z – zol < R, and has a zero
Res
20 = m.
(b)
a pole of order n at zo, show that
Suppose that f(2) is analytic in 0< z- zol < R, and has
Res
-п.
(c)
in a domain D, which does not pass through any pole or zero of f
and encloses its interior region Nc D. Suppose that in 2 there are:
• a pole z1 of order 3, a pole z2 of order 8, a pole z3 of order 1 of
the function f;
• a zero wi of order 2, a zero wz of order 4 of the function f.
Suppose that is a positively oriented simple closed curve
1
Compute
2ni
f'(2)
-dz.
f(2)
Hint You might want to use the residue theorem and results from
previous sub-questions to solve this sub-question.
2.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
