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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
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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[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

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: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.
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.
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.
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