# MATH30040 Complex Analysis

Complex numbers were introduced by mathematicians working in the 16th century. Complex analysis, which first emerged in the 19th century largely due to the pioneering work of Augustin-Louis Cauchy, is a branch of mathematical analysis that focuses on differentiable (more precisely 'analytic' or 'holomorphic') complex-valued functions defined on the set of complex numbers and certain subsets of it. As well as being a rich and beautiful theory in its own right, complex analysis has innumerable applications in physics, engineering and other areas of mathematics such as analytic number theory. We will cover topics chosen from the following list:

1. differentiation on the complex plane; exponential, trigonometric and logarithmic functions;
2. path integration; Cauchy's Theorem and Cauchy's Integral Formula;
3. power series; Taylor's Theorem and its consequences;
4. Laurent series; classification of singularities; Cauchy's Residue Theorem, and
5. applications of the Residue Theorem, such as Fourier Transforms and the evaluation of real integrals and infinite sums.

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Curricular information is subject to change

Learning Outcomes:

On successful completion of this module, the student is expected to be able to understand the definitions, theorems and examples covered in all of the topics listed above, and carry out associated computations. These include, but are not limited to, stating the main theorems in the module, together with some of their consequences; prove a selection of these results; apply the Cauchy-Riemann equations and Cauchy's Integral Formula; evaluate the integral of a complex function along a path; compute Taylor and Laurent expansions of various functions; compute residues and contour integrals; and evaluate Fourier Transforms and real integrals and infinite sums using residues.

Student Effort Hours:
Student Effort Type Hours
Lectures

30

Tutorial

10

Specified Learning Activities

24

Autonomous Student Learning

46

Total

110

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

Prior knowledge of complex numbers, differentiation of functions of one variable, partial derivatives, line/path integration, continuity of functions of one variable, and infinite series will be assumed.

In particular, the student should have taken a level one module on calculus of one variable and a level two module on calculus of several variables.

Module Requisites and Incompatibles
Incompatibles:
MST30050 - Complex Analysis

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Class Test: Two 50-minute written class tests, worth 10% each. Throughout the Trimester n/a Standard conversion grade scale 40% No

20

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

80

No

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour