2023/24 Undergraduate Module Catalogue

MATH3017 Calculus in the Complex Plane

15 creditsClass Size: 95

Module manager: Dr Joao Faria Martins; Sebastian Eterovic
Email: J.FariaMartins@leeds.ac.uk;S.Eterovic@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2023/24

Pre-requisites

MATH2017

Real Analysis

This module is mutually exclusive with

MATH2016	Analysis
MATH2090	Real and Complex Analysis

This module is not approved as a discovery module

Module summary

Calculus becomes immensely more powerful and, in many ways, simpler, if one allows the functions and variables under consideration to take values in the complex plane, rather than restricting them to the real line. This module will develop the theory of differentiable functions of a single complexvariable, an outstanding highlight of 19th century mathematics, in a coherent and mathematically rigorousway. Towards the end, complex analytic techniques will be used to solve seemingly intractable problems inreal analysis (exact computation of integrals over the real line, and exact summation of series, for example).

Objectives

a) To develop a mathematically rigorous and coherent theory of the calculus of holomorphic functions on
the complex plane.
b) To compare and contrast this theory with the theory of real differentiable functions.
c) To apply this theory to otherwise intractable problems in real analysis.

Learning outcomes
On completion of this module, students should be able to:
a) use the Cauchy-Riemann equations to decide whether a given function is holomorphic;
b) construct conjugate pairs of harmonic functions;
c) compute contour integrals, from first principles, using the fundamental theorem of the calculus, Cauchy's
theorem and Cauchy's integral formula;
d) compute the Laurent series of a holomorphic function about an isolated singularity;
e) classify the singularities of holomorphic functions and to compute, in the case of a pole, its order and
residue;
f) evaluate typical definite integrals by using the calculus of residues; apply this technique to solve problems
in real analysis;
g) construct rigorous proofs of (a selection of) the theorems presented.

Syllabus

a) Complex differentiability, the Cauchy-Riemann equations, relation to harmonic functions.
b) Contour integration, elementary methods of evaluation, the Fundamental Theorem of the Calculus, the Estimation Lemma.
c) Cauchy's Theorem (proof deferred), Cauchy's Integral Formula, applications of this (Liouville's Theorem, the Maximum Modulus Principle, the Fundamental Theorem of Algebra).
d) The Weierstrass M-test, Taylor series of holomorphic functions, holomorphic implies analytic, contrast with real analysis.
e) Laurent's Theorem, classification of singularities of holomorphic functions.
f) Cauchy's Residue Formula. Applications, including the argument principle and Rouché's theorem.
g) Applications of contour integration in real analysis.
h) Proof of Cauchy's Theorem.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	33	1.00	33.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Opportunities for Formative Feedback

6 problem sheets, self-assessed, each supported by a workshop.

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr 30 mins	100.00
Total percentage (Assessment Exams)		100.00

Please note: the exact length of the exam is to be determined, and is subject to change.

Reading list

The reading list is available from the Library website

Last updated: 14/09/2023

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