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.

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

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

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

• Feedback individually to students, post-assessment

Homeworks will be graded and returned. Midterms will be graded and students will have the option to view their scripts