This module is devoted to the calculus of functions of a complex variable, that is, to functions whose domain and range are regions of the complex plane rather than subsets of the real line. A grounding in functions of a complex variable is provided and the interplay between analytic and geometric factors in complex function theory is demonstrated. The module aims to develop the manipulative and reasoning skill of each student in this elegant and useful area of mathematics.
Review of complex numbers and their graphical representation. Manipulation of inequalities. Factorisation of complex polynomials. Contours, simple closed curves, open and connected subsets in the complex plane.
Analytic Functions
Functions of a complex variable, real and imaginary parts, differentiability. Analytic functions and the Cauchy-Riemann conditions. Laplace’s equation: harmonic and conjugate harmonic functions. Polynomials, exponential, trigonometric, hyperbolic and logarithmic functions.
Complex Integration
Contour integrals, the Fundamental Theorem of Calculus, Cauchy’s integral Theorem and Integral Formulae. Morera’s theorem, Liouville’s theorem and the Fundamental Theorem of Algebra.
Taylor and Laurent Series
Sequences, series and convergence in the complex plane. Power series: Taylor and Laurent series, uniform convergence of series. Classification of singularities and zeros. The Residue theorem and applications.
Algebra and Geometry of the Complex Plane
Review of complex numbers and their graphical representation. Manipulation of inequalities. Factorisation of complex polynomials. Contours, simple closed curves, open and connected subsets in the complex plane.
Analytic Functions
Functions of a complex variable, real and imaginary parts, differentiability. Analytic functions and the Cauchy-Riemann conditions. Laplace’s equation: harmonic and conjugate harmonic functions. Polynomials, exponential, trigonometric, hyperbolic and logarithmic functions.
Complex Integration
Contour integrals, the Fundamental Theorem of Calculus, Cauchy’s integral Theorem and Integral Formulae. Morera’s theorem, Liouville’s theorem and the Fundamental Theorem of Algebra.
Taylor and Laurent Series
Sequences, series and convergence in the complex plane. Power series: Taylor and Laurent series, uniform convergence of series. Classification of singularities and zeros. The Residue theorem and applications.
Lectures and tutorials
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Formal Examination | 70 |
| Other Assessment(s) | 30 |