# MST30050 Complex Analysis

The main topics taught in the course are as follows: an introduction to complex numbers, regions in the complex plane, complex functions, the complex logarithm function, the derivative of a complex function, the Cauchy-Riemann equations, harmonic functions, holomorphic functions and their properties, complex integrals, Cauchy's Integral Theorem, Cauchy's Integral Formulae, Taylor and Laurent series, the calculus of residues, Cauchy's Residue Theorem. Rigorous proofs of some of the theorems in this module, such as Cauchy's Integral Theorem, belong to a more advanced course but, where possible, weaker versions of such theorems are proved.

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Curricular information is subject to change

Learning Outcomes:

A student who successfully completes this module should be able to perform any of the algorithms taught in the module, for example, using the Cauchy-Riemann equations to decide whether or not a function is holomorphic, constructing the second function of a pair of harmonic conjugates, evaluating a complex integral by parametrising its path, using one or more of Cauchy's theorems to evaluate a complex integral. A student must understand the theory behind such techniques and be able to demonstrate that understanding.

More generally students should be able to do the following:

WRITE MATHEMATICS: Students should be able to recognise, read and correctly use standard mathematical symbols and notation, to correctly write a mathematical statement and to recognise when such a statement is not correctly written.

QUESTION: Students should be able to ask pertinent questions themselves, to decide which questions are most relevant, which questions are answerable, which questions they should start with, etc.

UNDERSTAND: Students must be able to understand the reasoning behind any methods or procedures they use and be able to demonstrate that understanding.

PRODUCE EXAMPLES: Students must be able to produce examples themselves, to illustrate a definition, to show a method, to test boundaries of an idea.

Indicative Module Content:

Elementary Properties of Complex Numbers
1.1 Origins of Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Review of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Operations with complex numbers . . . . . . . . . . . . . . . . . 6
1.2.2 The Argand Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 Complex Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.6 Polar form of a complex number . . . . . . . . . . . . . . . . . . . 11
1.2.7 Calculating the Argument . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.8 Properties of the Modulus . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Multiplication in Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 De Moivre’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 The complex numbers cannot be ordered

2 Functions of a Complex Variable. . . . . . . . . . . . . . . . . 23
2.1 Topology of the Complex Numbers. . . . . . . . . . . . . . . . . . . . . 24
2.2 Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2 Translations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
12.2.3 Rotations and dilations/contractions . . . . . . . . . . . . . . . . 34
2.2.4 Squaring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.5 Compositions of Fundamental Functions . . . . . . . . . . . . . . 37
2.3 Limits of Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Differentation of Complex Functions
3.1 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 The Cauchy Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Complex Power Series
4.1 Complex Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Series of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 The Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2 The Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.3 The Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Radius of convergence . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2 The complex exponential and trigonometric functions . . . . . . 64
4.3.3 Termwise differentiation of power series . . . . . . . . . . . . . . 66
4.3.4 The complex hyperbolic cosine and sine functions . . . . . . . . 68
4.4 The Complex Logarithmic Function . . . . . . . . . . . . . . . . . . . . . 70

5 Complex Integration
5.1 Paths in the Complex Plane. . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Complex line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
25.3 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Laurent’s Series and Residues
6.1 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.1 Removable Singularities . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.2 Poles of Finite Order . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.3 Essential Singularities . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Residues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3.1 Calculating Residues . . . . . . . . . . . . . . . . . . . . . . . . . 105

Student Effort Hours:
Student Effort Type Hours
Lectures

18

Tutorial

10

Specified Learning Activities

36

Autonomous Student Learning

34

Total

98

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning..
Requirements, Exclusions and Recommendations
Learning Requirements:

MST20040 or equivalent. (For equivalence , check with the module co-ordinator.)

Learning Recommendations:

MST20040 or equivalent. (For equivalence , check with the module co-ordinator.)

Module Requisites and Incompatibles
Incompatibles:
MATH30040 - Functions One Complex Variable

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Assignment: Homework assignment Throughout the Trimester n/a Standard conversion grade scale 40% No

15

Examination: Examination 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

70

Continuous Assessment: Mid-Term Test Week 7 n/a Standard conversion grade scale 40% No

15

Carry forward of passed components
No

Resit In Terminal Exam
Spring Yes - 2 Hour