Module Descriptors

Mathematical Studies

MA302: Complex Variables (5 ECTS)

This course introduces complex variable theory. Topics covered include: Cauchy-Riemann equations, Laplace's equation, complex numbers to the power of complex numbers, Integral evaluation in the complex plane, Cauchy's integral theorem, Cauchy's integral formula and Cauchy's integral formulae for derivatives, residues and the residue theorem.

Taught in Semester(s) 2. Examined in Semester(s) 2.

Workload: 106 hours (24 Lecture hours, 12 Tutorial hours, 70 Self study hours).

Module Learning Outcomes. On successful completion of this module the learner should be able to:

  1. Simplify complex numbers and plot the result in the Argand diagram. Calculate derivatives of a complex function. Define: complex conjugate, real part and imaginary part of a complex number.
  2. Use the Cauchy-Riemann equations to find the points in the complex plane where a function is differentiable. Compute the derivative at these points.
  3. Show that certain functions are harmonic functions and calculate the harmonic conjugate of a harmonic function.
  4. Write complex numbers in polar form; find and plot their roots in the complex plane. Find complex powers of complex numbers and write the result in polar form or in the form: $a+ib$. Verify expressions for various inverse trigonometrical functions.
  5. State Cauchy's integral theorem and all the associated technical details. Compute integrals of analytic and non-analytic functions over various paths in the complex plane.
  6. State Cauchy's integral formula and Cauchy's integral formula for derivatives. Use these to compute integrals around simple closed curves where there are poles within these simple closed curves.
  7. Obtain the Taylor series centered about a point. Find the Laurent series centered about a point valid in different regions.
  8. State the Residue Theorem. Use it to compute integrals around simple closed curves.

Indicative Content

This course introduces the theory of complex variables. The material covered includes:

  1. Complex functions: simplification of complex functions, terminology used in connection with complex numbers, differentiation of complex functions, Cauchy-Riemann equations, evaluation of derivatives, Laplace's equation, harmonic functions, harmonic conjugate.
  2. Powers of complex numbers: polar form, roots of complex numbers, complex powers of complex numbers, plotting in the Argand diagram, inverse trigonometrical functions and their evaluation.
  3. Integral evaluation: parameterisation, analytical and non-analytical integrands, Cauchy's integral theorem.
  4. Integral evaluation: Cauchy's integral formula and Cauchy's integral formula for derivatives.
  5. Residues: classification of singularities, computation of residues, residue theomem, computation of integrals using the residue theorem.

Module Resources

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