## Module and Programme Catalogue

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## MATH2016 Analysis

### 15 creditsClass Size: 220

Module manager: Professor Martin Speight
Email: J.M.Speight@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2016/17

### Pre-requisite qualifications

(MATH1010 and MATH1026) or (MATH1050 and MATH1055) or equivalent

### This module is mutually exclusive with

 MATH2015 Analysis 2 MATH2090 Real and Complex Analysis

This module is approved as a discovery module

### Module summary

This module aims to develop the ideas of continuity, differentiability and integrability and in particular show how they can be extended to complex valued functions. It will develop students' ability to appreciate the importance of proofs, and to understand and write them.

### Objectives

On completion of this module, students should be able to:
a) use the epsilon-delta formulation to show that a function is continuous;
b) calculate upper and lower Riemann sums;
c) apply convergence tests to series of real and complex numbers and evaluate the radius of convergence of power series;
d) use the Cauchy-Riemann equations to decide where a given function is analytic;
e) compute standard contour integrals using the fundamental theorem of the calculus, Cauchy's theorem or Cauchy's integral formula;
f) classify the singularities of analytic functions and to compute, in the case of a pole, its order and residue;
g) evaluate typical definite integrals by using the calculus of residues.

### Syllabus

1. Epsilon-delta definition of continuity for a function of a real variable.
2. Riemann integration for real valued functions. Formal properties of the integral. The Fundamental Theorem of the Calculus.
3. Basic ideas of complex function theory. Limits, continuity, analytic functions, Cauchy-Riemann equations.
4. Contour integrals. Cauchy's theorem, Cauchy's integral formula.
5. Power series. Analytic functions represented as Taylor or Laurent series. Singularities. Orders of poles, Cauchy's residue theorem, evaluation of definite integrals.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 33 1.00 33.00 Private study hours 107.00 Total Contact hours 43.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying notes between lectures: 53 hours
Doing problems: 40 hours
Exam preparation: 14 hours

### Opportunities for Formative Feedback

Regular problems sheets

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Written Work * 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 85.00 Total percentage (Assessment Exams) 85.00

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