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## MATH2015 Analysis 2

### 20 creditsClass Size: 180

Module manager: Professor J Wood (Sem 1) Dr K Houston (Sem 2)
Email: j.c.wood@leeds.ac.uk

Taught: Semesters 1 & 2 (Sep to Jun) View Timetable

Year running 2011/12

### Pre-requisites

 MATH1035 Analysis

### This module is mutually exclusive with

 MATH2090 Real and Complex Analysis

This module is approved as an Elective

### Module summary

This module aims to develop the ideas of MATH1035 Analysis and show how they can be extended to the 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) apply convergence tests to series of real and complex numbers and evaluate the radius of convergence of power series;
b) use the Cauchy-Riemann equations to decide where a given function is analytic;
c) calculate upper and lower Riemann sums;
d) determine whether improper integrals converge;
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. Revision of complex numbers up to the complex exponential function.
2. Sequences of complex numbers. Limits. Convergent sequences are bounded. Cauchy sequences. Completeness of C.
3. Convergent of complex series. Revision of convergence tests from Analysis 1. Power series and radius of convergence.
4. Open sets. Continuous complex valued functions.
5. Differentiability of complex functions. Rules for derivatives. Cauchy-Riemann equations.
6. Harmonic Functions. Conformal transformations.
7. Riemann integration for real valued functions. Formal properties of the integral.
8. The Fundamental Theorem of the Calculus.
9. Improper integrals.
10. Uniform convergence and uniform continuity.
11. Contour integration. Definitions of contours and closed contours. Integrals of continuous functions along a contour.
12. Estimates for integrals. Fundamental theorem of the calculus for analytic functions.
13. Cauchy's theorem and integral formula. Winding number. Cauchy's theorem Cauchy's integral formula. Liouville's theorem.
14. Taylor's theorem. Formula for coefficients in complex Taylor series. Differentiable functions are infinitely differentiable.
15. Calculus of residues. Definitions of pole of order m, simple pole, removable singularity, essential singularity, residue.
16. Cauchy's residue theorem. Application to calculation of definite integrals.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 20 1.00 20.00 Lecture 44 1.00 44.00 Private study hours 136.00 Total Contact hours 64.00 Total hours (100hr per 10 credits) 200.00

### Opportunities for Formative Feedback

Regular problems sheets

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Written Work coursework + in-course tests 20.00 Total percentage (Assessment Coursework) 20.00

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

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 00 mins 80.00 Total percentage (Assessment Exams) 80.00

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