Edinburgh Mathematics Programme

Level 3 Module

321 CAn Complex Analysis (S1)

Introduction

Complex analysis is a central area of mathematics which is concerned with differentiable functions of a complex variable. One of the main techniques used in this study is that of line integrals, that is, integrals of functions defined on the complex plane over paths in the plane. It turns out that it is possible to evaluate various ordinary real integrals (that would otherwise be very difficult to calculate) using this idea. The subject has close links with analytic number theory, differential equations, Fourier series and analysis, geometry and the actions of groups on the plane, and has numerous applications across all of mathematics. This course covers the basic fundamental results of the subject.

Aims

  1. To study basic properties of complex differentiable functions.
  2. To relate complex differentiable functions with line integrals along closed curves in the plane.
  3. To discuss Cauchy's Theorem, Cauchy's Integral Formula and their basic applications.
  4. To introduce Taylor and Laurent series.
  5. To use residues to evaluate certain real integrals which cannot be handled by the standard methods of calculus.

Prerequisites

  1. MS0095 FoC
  2. MS0097 SVC

Syllabus summary

Differentiability of complex functions, Cauchy-Riemann equations. Complex line integrals. Cauchy's Theorem for a disc, Cauchy's Integral Theorem. Power series expansions, identity theorem and Liouville's Theorem. Laurent's Theorem, the Residue Theorem and applications to the evaluation of standard real integrals.

Text

J. E. Marsden and M. J. Hoffman, Basic complex analysis, Freeman, QA 331.7 Mar. This book is at about the right level (but covers more than is in this course).

There are many uses of complex variables in applied mathematics; several are discussed in a quick and clear manner in "Introduction to Applied Mathematics" by G. Strang (Wellesley Cambridge Press) QA 37.2 Str, pages 330-366. The book is in the reserve sections of the JCMB and Main University Libraries.

Syllabus

  1. Introduction and revision: Review of basic algebra and analysis in C, power series, exponential and logarithm, nth roots. (3)
  2. Complex differentiability: Cauchy-Riemann equations, domains. Real and imaginary parts of complex differentiable functions are harmonic. (2)
  3. Curves and integration: Parametrisation and contour integration [done by the substitution method not Riemann sums], integral of a derivative depends on end points of the curve not curve, f'(z) = 0 in U implies f is constant in U, the M-L Lemma [modulus of integral is no more than max-modulus on curve times length of curve]. Maximum Principle. (4)
  4. Cauchy's Theorem and integral formula: A fairly simple form of Cauchy's Theorem (proved using either Green's Theorem or sketched using the triangulation method), Cauchy's integral theorem and evaluation of some examples. (4)
  5. Power series expansions: Taylor series, Cauchy estimates, Liouville's theorem and the Fundamental Theorem of Algebra [or do both in 4], zeros, identity theorem. (4)
  6. Laurent's Theorem: Sketch of proof of Laurent's Theorem, poles and residues, and calculation of residues for poles of order one and two with examples. (4)
  7. Residue Theorem: A sketch of a proof of the Residue Theorem, applications to standard examples of real integrals. (3)

Notes and Links

  1. It is essential to revise complex numbers briefly at the start of the module so that trivial problems here are not a block to later understanding.
  2. Creating links with other modules is important where it can be done easily -- such as in the link between Taylor/Laurent Series and Fourier Series.
  3. There should be proofs but no undue emphasis on epsilon-delta; this is a course for all. Current experience indicates that Green's theorem is not remembered so it must be fully revised if used.
  4. Examples need to be done at all stages, even some that can be handled more easily by general theorems later.
  5. There is scope for Maple use in evaluating residues.
  6. Maximum Principle is another good link topic with PDE's.
  7. More challenging problems should be included on the problem sheets after the more routine ones. In this way it may be possible, for instance, to introduce special cases of conformal mapping (not on the syllabus).

Outcomes

  1. The students should be able to understand analytic functions and their elementary properties.
  2. They should know the Cauchy-Riemann equations and that real and imaginary parts of analytic functions are harmonic.
  3. They should be able to perform direct calculation of line integrals in simple cases.
  4. They should have familiarity and facility with Taylor and Laurent series and with the calculation of residues.
  5. They should be able to use Cauchy's Theorem, the Cauchy Integral Theorem and the Residue Theorem in basic situations and to evaluate standard real integrals.

  • CJS 6.3.02