MA3RCA-Real and Complex Analysis

Module Provider: Mathematics and Statistics
Number of credits: 20 [10 ECTS credits]
Terms in which taught: Autumn / Spring term module
Pre-requisites: MA1FM Foundations of Mathematics and MA1CA Calculus and MA1LA Linear Algebra and MA2RA1 Real Analysis I
Non-modular pre-requisites:
Modules excluded: MA2RA2 Real Analysis II or MA2RAX Real Analysis or MA2CA1 Complex Analysis I or MA3CA1 Complex Analysis I or MA2RCA Real and Complex Analysis
Module version for: 2017/8

Module Convenor: Prof Michael Levitin


Summary module description:
The first part of this module continues the study of analysis to the point where it relates to topics in other courses, such as integration and differentiation. The second part provides an introduction to complex analysis.

To further deepen students’ understanding of real analysis and introduce students to complex analysis. To enable students to relate this to calculus and demonstrate that analysis can produce remarkable results that calculus cannot, and enable them to use complex variable techniques, particularly in some cases where the original problem does not involve complex numbers.

Assessable learning outcomes:

By the end of the module, students are expected to be able to:

• use the concepts of continuity, differentiation and integration in a rigorous way;

• manipulate infinite series and apply them in problems involving continuity, differentiation and integration;

• solve problems involving holomorphic functions;

• recognize and be able to apply the complex exponential and logarithm;

• evaluate path integrals of complex functions;

• identify singularities and residues of holomorphic functions;

• calculate appropriate real integrals using complex techniques.

Additional outcomes:
By the end of the module the student will begin to understand and recognise some of the structure of holomorphic functions.

Outline content:
The definition of the Riemann integral and Riemann sums. The properties of the integral. Integrability of continuous functions. Fundamental theorem of calculus. Sequences and series of functions. Uniform convergence. Power series. Special functions, such as exponential, logarithmic and trigonometric functions. Applications of real analysis.

Differentiable functions of a complex variable are remarkably well-behaved, and most of the technical complications of the real case do not arise with complex functions. This leads to some remarkably powerful results, and it turns out that complex variable techniques often offer the simplest method of evaluating certain real integrals. The notion of complex differentiability relates closely with power series. Contour integration in the complex plane will be introduced and the remarkable theorem of Cauchy established, from which a whole range of applications follow. Some applications to the evaluation of real integrals are given.

Brief description of teaching and learning methods:
Lectures, supported by problem sheets and lecture-based tutorials.

Contact hours:
  Autumn Spring Summer
Lectures 20 20
Tutorials 10 10
Guided independent study 70 70
Total hours by term 100.00 100.00
Total hours for module 200.00

Summative Assessment Methods:
Method Percentage
Written exam 80
Set exercise 20

Other information on summative assessment:

Assignments and one examination (level 6).

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convenor will apply the following penalties for work submitted late, in accordance with the University policy.

  • where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
  • where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

  • The University policy statement on penalties for late submission can be found at:
    You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

    Length of examination:

    3 hours.

    Requirements for a pass:

    A mark of 40% overall.

    Reassessment arrangements:
    One examination paper of 3 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (80% exam, 20% coursework).

    Additional Costs (specified where applicable):
    1) Required text books:
    2) Specialist equipment or materials:
    3) Specialist clothing, footwear or headgear:
    4) Printing and binding:
    5) Computers and devices with a particular specification:
    6) Travel, accommodation and subsistence:

    Last updated: 24 July 2017

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