MAT3705
ASSIGNMENT 4 2023
, MAT3705 - 2023
Assignment 04
Chapters 6 and 7
Due date: 24 August 2023
Please note:
No typed solutions will be accepted. Only handwritten assignments will be accepted. Please scan your
solutions and upload them as a single pdf on the mymodules page.
Please make sure that you submit the correct file on the MAT3705 mymodules page. No changes will be
allowed after the closing date. No e-mailed solutions will be accepted.
Your submission has to be your own work. Copying answers, using answers from the internet, buying
solutions, etc. are academic offences and detrimental to your own learning.
You have three weeks to submit your solutions.For this reason, loadshedding and technical difficulties will
not be considered valid excuses for failing to submit.
This assignment has 2 pages and 6 questions.Please make sure that you answer all questions.
Questions:
1. Let f (z) = cos (π/z).
(a) Write down the Laurent series of f about the point z = 0 and specify the region of validity.
(b) Describe why z = 0 is an isolated singularity of f .
(c) What type of isolated singularity is z = 0?
(d) Use the definition of a residue to calculate the residue of f at the point z = 0.
z+2
2. Let g(z) = .
(z 2 + 1)(z − i)
(a) Locate and classify all the isolated singularities of g and compute the residues at each of these singu-
larities. Provide motivations for your answers.
(b) Let C denote the positively oriented contour C = {z ∈ C : |z − i| = 3}. Use Cauchy’s Residue
R
Theorem to calculate C g(z) dz.
1
ASSIGNMENT 4 2023
, MAT3705 - 2023
Assignment 04
Chapters 6 and 7
Due date: 24 August 2023
Please note:
No typed solutions will be accepted. Only handwritten assignments will be accepted. Please scan your
solutions and upload them as a single pdf on the mymodules page.
Please make sure that you submit the correct file on the MAT3705 mymodules page. No changes will be
allowed after the closing date. No e-mailed solutions will be accepted.
Your submission has to be your own work. Copying answers, using answers from the internet, buying
solutions, etc. are academic offences and detrimental to your own learning.
You have three weeks to submit your solutions.For this reason, loadshedding and technical difficulties will
not be considered valid excuses for failing to submit.
This assignment has 2 pages and 6 questions.Please make sure that you answer all questions.
Questions:
1. Let f (z) = cos (π/z).
(a) Write down the Laurent series of f about the point z = 0 and specify the region of validity.
(b) Describe why z = 0 is an isolated singularity of f .
(c) What type of isolated singularity is z = 0?
(d) Use the definition of a residue to calculate the residue of f at the point z = 0.
z+2
2. Let g(z) = .
(z 2 + 1)(z − i)
(a) Locate and classify all the isolated singularities of g and compute the residues at each of these singu-
larities. Provide motivations for your answers.
(b) Let C denote the positively oriented contour C = {z ∈ C : |z − i| = 3}. Use Cauchy’s Residue
R
Theorem to calculate C g(z) dz.
1
