Complex & Asymptotic Methods - MAT00048H

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  • Department: Mathematics
  • Module co-ordinator: Prof. Evgeny Sklyanin
  • Credit value: 10 credits
  • Credit level: H
  • Academic year of delivery: 2018-19

Related modules

Co-requisite modules

  • None

Module will run

Occurrence Teaching cycle
A Autumn Term 2018-19

Module aims

This module further develops Complex Analytic methods, and provides techniques that can be used to evaluate nontrivial integrals of functions with branch cuts. It also introduces and develops asymptotic methods which give useful estimates of the growth of functions and can also be used to give accurate estimates of various integrals. These techniques are used in various areas of pure and applied mathematics. Some “special functions” such as the Gamma and Beta functions are studied in detail using the methods of the module.

Module learning outcomes

  • Be able to confidently apply tools and techniques of complex analysis in a variety of problems, including evaluation of contour integrals and solving differential equations.
  • Know various properties of the Gamma, Beta and other special functions.
  • Be able to find asymptotic expansions for a variety of functions.

Module content

[Pre-requisites for Natural Sciences students: must have taken Maths for Sciences III MAT00019I.]

Syllabus

  • Residue calculus, stressing practical computations. Classification of singularities (removable, pole, essential). Calculation of residues of higher order poles. Contour integrals. Using Cauchy’s theorem to evaluate contour integrals. Laurent series, residue at infinity. Examples, including evaluation of nontrivial Fourier and Laplace transforms.

  • Improper integrals. Convergence and analyticity criteria. Analytic continuation. Examples: Gamma and Beta functions

  • Multivalued functions, branch points. Notion of Riemann surface, cuts. Practical evaluation of contour integrals on the complex plane with cuts. Examples, including Gamma and Beta functions

  • Elements of asymptotic analysis. The symbols O, o , ~. Asymptotic expansions. Laplace’s (stationary point) method. Examples, including Stirling’s formula for the Gamma function; Airy’s function.

Academic and graduate skills

  • Academic skills: the techniques taught are used in many areas of pure and applied mathematics.

  • Graduate skills: through lectures, examples, classes, students will develop their ability to assimilate, process and engage with new material quickly and efficiently. They develop problem solving-skills and learn how to apply techniques to unseen problems.

Assessment

Task Length % of module mark
University - closed examination
Complex & Asymptotic Methods
2 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
University - closed examination
Complex & Asymptotic Methods
2 hours 100

Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

  • H A Priestley, Introduction to Complex Analysis (S 7.55 PRI)
  • Ian Stewart and David Tall, Complex Analysis: The Hitchhiker's Guide to the Plane (S 7.55 STE)
  • M.J. Ablowitz and A.S. Fokas, Complex variables: Introduction and Applications (Cambridge University press)
  • G F Simmons, Differential Equations, with Applications and Historical Notes, Tata MacGraw-Hill (paperback) (S7.38 SIM)
  • E T Whittaker and G N Watson, A Course of Modern Analysis, (Cambridge University Press)



The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.