[Date]
MAT3705 Assignment 4 2024
- DUE 5 September 2024
QUESTIONS WITH ANSWERS
, 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.
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.
Question 1: Poles of Functions
MAT3705 Assignment 4 2024
- DUE 5 September 2024
QUESTIONS WITH ANSWERS
, 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.
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.
Question 1: Poles of Functions
