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MAT3705 Assignment 4 (COMPLETE ANSWERS) 2024 - DUE 5 September 2024
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, MAT3705 Assignment 4 (COMPLETE ANSWERS)2024 - DUE 5 September 2024 ; 100% TRUSTED
Complete, trusted solutions and explanations.
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. 6. Use Rouche’s Theorem to
determine the number of roots of h(z) = 3z3 + 2z2 + 2z − 8 = 0
inside the disc {z ∈ C : |z| < 2}. Provide reasons for your
answer.
2. 1. Poles of f(z)f(z)f(z) and g(z)g(z)g(z) at z=iz = iz=i
3. Given:
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