## MATH3017 Calculus in the Complex Plane

### 15 creditsClass Size: 50

Module manager: Professor Jonathan Partington
Email: J.R.Partington@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2019/20

### 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 allowsthe functions and variables under consideration to take values in the complex plane, rather than restrictingthem 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 RouchÃ©'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

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated