MAT3705 Assignment 4 2024 - DUE 5 September 2024

MAT3705 Assignment 4
2024 - DUE 5 September
2024
QUESTIONS WITH COMPLETE ANSWERS




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, MAT3705 Assignment 4 2024 - DUE 5 September 2024



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 4 . Calculate the value of Z 2π 0 1 + cos θ 5 + 3 sin θ
dθ.

4. Use Residue Theory to calculate Z ∞ −∞ x2 (x2 + 9)2 dx.

5. Let f(z) = z2 (z + 4)(z2 − 9) . Show that lim R→∞ Re Z CR f(z)ei5z dz
= 0, where CR denotes the positively oriented contour {Reiθ : 0 ≤ θ ≤
π}. Justify all steps.