# MA271-10 Mathematical Analysis 3

Department
Warwick Mathematics Institute
Level
Vedran Sohinger
Credit value
10
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

##### Introductory description

This is the third module in the series Analysis 1, 2, 3 that covers rigorous Analysis. It covers convergence of functions and its applications to Integration, Fourier Series and Complex Analysis.

##### Module aims
1. Continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions.
2. Fourier series and their convergence.
3. Foundations of Complex Analysis.
##### Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Uniform convergence of sequences and series of functions; Weierstrass M-test
• Application to integration: integrals of limits and series, differentiation under the integral sign
• Fourier series: convergence, Parseval, and Gibbs phenomenon (differentiability and rate of decay of coefficients)
• Complex power series and classical functions (exponential, logarithm, sine and cosine, including periodicity)
• Complex integration, contour integrals and Cauchy’s Theorem
• Applications of Cauchy’s formula to evaluate real integrals
• Laurent series, Calculus of residues
##### Learning outcomes

By the end of the module, students should be able to:

• Learn how to compute contour integrals: Cauchy's integral formulas and applications
• Understand uniform and pointwise convergence of functions together with properties of the limit function
• Learn the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions
• Develop working knowledge of complex differentiability (Cauchy-Riemann equations) and complex power series
• Develop understanding of Fourier Series including Gibbs phenomenon
##### Subject specific skills
• Working knowledge of series and sequences, including the development of the notions of convergence and uniform converge for sequences and series of functions.
• Good understanding of Fourier series, including their convergence, Parseval's identity and Gibbs phenomenon.
• Working knowledge of Complex Analysis, including power series, exponential and circular maps, contour integration.
• Mastery of applications of Cauchy's formula to compute integrals in R.
##### Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.

## Study time

Type Required
Lectures 20 sessions of 1 hour (20%)
Tutorials 9 sessions of 1 hour (9%)
Private study 71 hours (71%)
Total 100 hours
##### Private study description

71 hours private study, revision for exams, and assignments

## Costs

No further costs have been identified for this module.

You do not need to pass all assessment components to pass the module.

##### Assessment group D
Weighting Study time
Assignment 15%
Examination 85%
##### Assessment group R
Weighting Study time
In-person Examination - Resit 100%
##### Feedback on assessment

Support classes, marked assignments and exam feedback.

## Courses

This module is Core for:

• Year 2 of UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
• UPXA-GF13 Undergraduate Mathematics and Physics (BSc)
• Year 2 of GF13 Mathematics and Physics
• Year 2 of GF13 Mathematics and Physics
• UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)
• Year 2 of FG31 Mathematics and Physics (MMathPhys)
• Year 2 of FG31 Mathematics and Physics (MMathPhys)

This module is Optional for:

• Year 3 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

This module is Core option list A for:

• UMAA-GV17 Undergraduate Mathematics and Philosophy
• Year 2 of GV17 Mathematics and Philosophy
• Year 2 of GV17 Mathematics and Philosophy
• Year 2 of GV17 Mathematics and Philosophy

This module is Option list A for:

• Year 2 of G302 Data Science
• Year 2 of G302 Data Science
• Year 2 of G4G1 Discrete Mathematics
• Year 2 of G4G1 Discrete Mathematics
• Year 2 of UCSA-G4G3 Undergraduate Discrete Mathematics
• USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
• Year 2 of GG14 Mathematics and Statistics
• Year 2 of GG14 Mathematics and Statistics
• USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics
• Year 2 of Y602 Mathematics,Operational Research,Stats,Economics
• Year 2 of Y602 Mathematics,Operational Research,Stats,Economics

This module is Option list B for:

• USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics
• Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
• Year 3 of Y602 Mathematics,Operational Research,Stats,Economics

This module is Option list E for:

• Year 3 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics