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Complex & Asymptotic Methods - MAT00078M

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  • Department: Mathematics
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23

Module summary

 

 

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations


Additional information

Pre-requisite information: MSc students should have taken a first course in complex analysis, going as far as the residue calculus.

Module will run

Occurrence Teaching period
A Autumn Term 2022-23

Module aims

This module develops Complex Analytic methods beyond the level of a first course in complex analysis and provides techniques that can be used to evaluate nontrivial integrals (including examples 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 functions and integrals. These techniques are used in many 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. Finally, the module develops techniques to solve differential equations by complex analytic means (integral representation).

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, and be able to derive and use, various properties of the Gamma, Beta and other special functions.
  • Be able to find asymptotic expansions for a variety of functions.
  • Be able to solve suitable differential equations by integral representation.

Module content

 

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 function.

  • Solving differential equations with contour integrals. Laplace method. Examples drawn from Airy functions, Hermite’s equation, the confluent hypergeometric equation, Bessel’s equation. (This topic is not taught in the H-level variant of this module.)

 

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. Students on this module will learn to work more independently and assimilate advanced material at a greater rate than those on the H-level variant.

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Module feedback

Current Department policy on feedback is available in the 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 constantly explores 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. In some instances it may be appropriate for the University to notify and consult with affected students about module changes in accordance with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.