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Durham University

Department of Mathematical Sciences

MATH2011 Complex Analysis II

The module develops the theory of functions of a single complex variable and gives an introduction to metric spaces. In your first year you will have already met several examples of complex functions such as polynomials in z; ℯz; cos(z) and sin(z). Functions of a complex variable behave in a very different way from their real counterparts. For example, a complex function which is differentiable once everywhere is differentiable any number of times.

Term 1

We begin by building up a library of complex functions, and investigating their geometrical properties. The crucial development in Term 1 is the generalisation to complex functions of the notion of real differentiation.

One example of the sort of geometry we discover is that complex differentiable functions are angle-preserving (or conformal) at all points in the plane where their complex derivative is non-zero. We also investigate the relation between complex differentiable functions and solutions to the Dirichlet problem - this latter being of great theoretical and physical importance.

Another topic is the study of Möbius transformations and their geometric properties. We further generalise continuity and convergence to more general metric spaces. This enables us to develop a unified approach to convergence in different settings. We finally consider different types of convergence for sequences and series of functions (e.g., power series), concentrating on (locally) uniform convergence because this has the nice properties you would hope were true!

Term 2

We investigate what happens when we integrate complex differentiable functions along curves in the complex plane. Integrating complex functions is surprisingly different from integrating real functions. Often it is much easier, and we will see how to evaluate real integrals using complex ones. The key for what follows is Cauchy's Theorem and Cauchy's integral formula.

We investigate the properties of complex differentiable functions and again we find more surprises. If f(z) is complex differentiable, then automatically is analytic, i.e., can be written locally as a power series; if f(z) is complex differentiable, then |f(z)| cannot have any local maxima; if f(z) is differentiable everywhere and bounded, then it must be constant. We shall also consider other applications and prove, for example, the fundamental theorem of algebra.

Outline of Course

Term 1

  • Complex differentiation (6 lectures): Open sets and convergence in the complex plane, complex functions, exponential & logarithm functions (cuts and branches), complex differentiation, Cauchy-Riemann equations.
  • Conformal mappings (7 lectures): Conformal maps, harmonic and holomorphic functions, Dirichlet boundary problem. Möbius transformations: group law, point at infinity, geometry, circles/lines to circles/lines, preservation of cross ratio, Möbius transformations of unit circle to itself, Riemann sphere and stereographic projection, statement of Riemann mapping theorem illustrated by further examples.
  • Metric spaces (8 lectures): Examples of metric spaces, open/closed sets, convergence, continuity, (path)-connectedness. (Sequentially) compactness, Heine-Borel theorem, compactness and continuity. Chordal metric on the Riemann sphere, hyperbolic metric on the unit disk, Möbius transformations as isometries, the hyperbolic plane of the Maths Building.
  • Uniform convergence (4 lectures): Pointwise and uniform convergence of sequences and series, locally uniform convergence and convergence in compact subsets, continuity of the limit, Weierstrass M-test for continuous functions. Power series: Review from Analysis I of disk of (locally uniform) convergence and ratio/root test, term by term differentiation and integration, Taylor series.

Term 2

  • Contour integrals (7 lectures): Curves in the complex plane, mention Jordan curve theorem, line/contour integrals in ℂ. Primitives and integrability, Cauchy's theorem, Cauchy's integral formula, (Cauchy-)Taylor theorem, Morera's theorem.
  • Fundamental theorems for holomorphic functions (6 lectures): Riemann extension theorem, Uniqueness theorem, Cauchy inequalities, Liouville's theorem, fundamental theorem of algebra, maximum principle, open mapping theorem (statement). Uniform convergence in compact subsets preserves holomorphicity, Weierstrass M-test for holomorphic functions, examples: Riemann zeta-function, Gamma-function. Winding numbers, analytic continuation, lemma of Schwarz, automorphisms of unit disk.
  • Singularities and meromorphic functions (6 lectures): Singularities, Casorati-Weierstrass, automorphisms of ℂ. Laurent's theorem, partial fraction theorem, residue theorem. Principal of the argument, Rouché's Theorem.
  • Calculus of residues, applications (4 lectures): Evaluation of integrals by calculus of residues, Fourier & Laplace transform with examples (Gaussian), Jordan's lemma, Poisson kernel as solution to the Dirichlet problem (sketch), partial fraction decomposition of cotangent, special values of Riemann zeta-function.

Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.

Reading List

Please see the Library Catalogue for the MATH2011 reading list.

Examination Information

For information about use of calculators and dictionaries in exams please see the Examination Information page in the Degree Programme Handbook.