MAT3705 Assignment 4 (COMPLETE ANSWERS) 2024 - DUE 5 September 2024 ; 100% TRUSTED Complete, trusted solutions and explanationsEnsure your success with us..

MAT3705
Assignment 4
(COMPLETE
ANSWERS)
2024 - DUE 5
September
2024
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Exam (elaborations)
MAT3705 Assignment 4 (COMPLETE ANSWERS) 2024 -
DUE 5 September 2024
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 Course
 Complex Analysis (MAT3705)
Institution
 University Of South Africa (Unisa)
 Book
 Complex Analysis

MAT3705 Assignment 4 (COMPLETE ANSWERS) 2024 - DUE 5 September
2024 ; 100% TRUSTED Complete, trusted solutions and explanations. For
assistance, Whats-App 0.6.7-1.7.1-1.7.3.9. Ensure your success with us.. 1.
Let f(z) = z2 (z−i)4 and g(z) = z2+1 (z−i)4 . Explain why f has a pole of order 4
at z = i, but g has a pole of order 3 at z = i. 2. Let f(z) = sin z (z − π)2(z + π/2)
and let C denote the positively oriented contour C = {z = 4eiθ ∈ C : 0 ≤ θ ≤
2π}. (a) Identify the types of isolated singularities of f and calculate the
residues of f at these points. Provide reasons for your answers. (b) Use
Cauchy’s Residue Theorem to calculate Z C f(z) dz. 3. Let f(z) = (z + 1)2 z(z +
3i)(z + i/3) (a) What type of isolated singularity is z = −i/3 of the function f?
Provide reasons for your answer. (b) Calculate Resz=−i/3f(z). 1 (c) Calculate
the value of k such that Z 2π 0 1 + cos θ 5 + 3 sin θ dθ = k Z C f(z) dz, where
C is the positively oriented contour C = {z = eit : 0 ≤ t ≤ 2π}. (d) You are told
(and do not have to calculate) that Resz=0f(z) = −1 and Resz=−3if(z) = 12+3i