## Module Content

The module covers
• Topic 1 (8 lectures),
• "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."
• Topic 2 (8 lectures),
• "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. Integral evaluation: parameterisation, analytical and non-analytical integrands, Cauchy's integral theorem."
• Topic 3 (8 lectures),
• "Integral evaluation: Cauchy's integral formula and Cauchy's integral formula for derivatives. Residues: classification of singularities, computation of residues, residue theorem, computation of integrals using the residue theorem." - and explains how these topics can be applied.
• Application 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. Use the Cauchy-Riemann equations to find the points in the complex plane where a function is differentiable. Compute the derivative at these points. Show that certain functions are harmonic functions and calculate the harmonic conjugate of a harmonic function."
• Application 2,
• "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. 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."
• Application 3.
• Application 3: "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. Obtain the Taylor series centered about a point. Find the Laurent series centered about a point valid in different regions. State the Residue Theorem. Use it to compute integrals around simple closed curves."

## Module Coordinates

• Lecturer: Dr Michael Hayes
• Lectures: Monday 12.00-12.50 in IT204 and Wednesday 12.00-12.50 in ADB-1020
• Tutorials: Wednesday 2.00-2.50 in ADB1020
• Recomended text: "Complex variables and applications, James Ward Brown & Ruel V. Churchill"
• Problem sheet: available here.
• Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard MA302 Complex Variables pages. Blackboard will also be used for announcements and for posting grades.

## Module Assessment

• End of semester examination: 50%.
• Continuous assessment: 30%.
• Communications skils: 20%.

## Supplementary Material and News

A model paper is avaialble here.

Clicker opinion polling may be used in some lectures.

## Lecture Notes

 Lecture Notes (click on number) Lecture Summaries 1 Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24