MAT3705 Assignment 3 Due 23 July 2026

UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology


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MAT3705: Complex Analysis

Assignment 3 — Learning Units 5 and 6, 2026

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MAT3705
Module Code:
Complex Analysis
Module Name:
Cauchy’s Integral Formula, Contour Inte-
Assignment Topic:
grals and Laurent Series (Learning Units 5
and 6)
Assignment 3
Assignment Number:
23 July 2026
Due Date:




Submitted in partial fulfilment of the requirements for MAT3705 — UNISA 2026

, MAT3705 - 2026

Assignment 3

Learning units 5 and 6

Due date: 23 July 2026



Please note:

ˆ Only handwritten assignments will be accepted. No typed solutions will be accepted.

ˆ 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.

ˆ Your submission has to be your own work. Using artificial intelligence, copying answers, using answers

from the internet, buying solutions, etc. are academic offences and detrimental to your own learning.

ˆ This assignment has 4 pages and 5 questions. Please make sure that you answer all the questions.

Questions:



Section A: Multiple choice



Please insert and complete the following table to answer the questions in Section A:



Question 1(a) 1(b) 1(c) 1(d)
Answer


e3z
1. Let f (z) = and let C = {z ∈ C : |z − i| = 2} denote a positively oriented contour which
(z 2 + 4)(z − 2i) Z
is traversed once. The aim of this question is to determine the value of f (z) dz using Cauchy’s Integral
C
Formula. It is advisable that you work through the problem in detail before completing the table above

(in other words, work through the problem as you normally would and then select the appropriate options

to enter into the table). Please note that only the completed table will be marked in this section.

(a) Select the correct option:

i. Both z = 2i and z = −2i are interior to C.

ii. Neither z = 2i nor z = −2i are interior to C.


1

, iii. z = 2i is interior to C and z = −2i is exterior to C.

iv. z = −2i is interior to C and z = 2i is exterior to C.

(b) Select the correct option:
e3z
i. If we let g(z) = , then g is analytic everywhere inside and on C.
(z − 2i)2
e3z
ii. If we let g(z) = , then g is analytic everywhere inside and on C.
z + 2i
e3z
iii. If we let g(z) = , then g is analytic everywhere inside and on C.
z − 2i
e3z
iv. If we let g(z) = 2 , then g is analytic everywhere inside and on C.
z +4
(c) Select the correct option:
e3z
Z
i. f (z) dz = 2πig(−2i), where g(z) = .
z − 2i
ZC
e3z
ii. f (z) dz = 2πig(2i), where g(z) = 2 .
C z +4
e3z
Z
iii. f (z) dz = 2πig ′ (−2i), where g(z) = .
C (z − 2i)2
e3z
Z
iv. f (z) dz = 2πig ′ (2i), where g(z) = .
C z + 2i
(d) Select the correct option:
πe6i (12 − i)
Z
i. f (z) dz = .
8
ZC
πe−6i (12 + i)
ii. f (z) dz = .
C 8
πe6i
Z
iii. f (z) dz = .
C 4
πe−6i
Z
iv. f (z) dz = .
C −2

Section B: Longer answers

Please make sure that you answer the following questions in full (by including all calculations and justifi-

cations).

2. Let γ(t) = −t + i(2t) for t ∈ [0, 1] and let f (z) = z 2 + z.

R
(a) Calculate γ f (z) dz, using Theorem 5.2.4 (see Lesson 24).
R
(b) Calculate γ f (z) dz, using Theorem 5.3.1 (see Lesson 25).

3
3. Let f (z) = . Calculate the Laurent series of f about the point z = −3i in the region 3 < |z + 3i|.
z(z + 3i)
z 2 ez
4. Find the first four non-zero terms in the Laurent expansion of f (z) = about the point z = 0 in the
sin z
region 0 < |z| < π, using division of known power series (see Lesson 30).



2

, UNISA | MAT3705 Assignment 3 — Complex Analysis



Question 1(a)

Let
e3z
f (z) =
(z 2 + 4)(z − 2i)

and let
C = {z ∈ C : |z − i| = 2}

be a positively oriented contour traversed once.

Select the correct option:

i. Both z = 2i and z = −2i are interior to C.
ii. Neither z = 2i nor z = −2i are interior to C.
iii. z = 2i is interior to C and z = −2i is exterior to C.
iv. z = −2i is interior to C and z = 2i is exterior to C.

The contour has centre
i = (0, 1)

and radius
2.


Distance from the centre to 2i:
|2i − i| = |i| = 1 < 2.

So, 2i is inside the contour.

Distance from the centre to −2i:


| − 2i − i| = | − 3i| = 3 > 2.


So, −2i is outside the contour.

Key Distinction
Answer: iii




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