The aim of the module is to extend the ideas developed in Mathematical Analysis I and II from real to complex numbers, to develop the theory of holomorphic functions, and to apply this theory to understand problems arising in real analysis or calculus.

At the end of this module, the learners will be able to:

- Define the field of complex numbers

- Determine continuity, differentiability, analyticity of a function and find the derivative of a function;

- Describe Complex integration and Cauchy’s theorem

- Explain Taylor and Laurent series

Skills

- Perform basic mathematical operations with complex numbers in Cartesian and polar forms;

- Work with functions (polynomials, reciprocals, exponential, trigonometric, hyperbolic,etc) of single complex variable and describe mappings in the complex plane;

- Work with multi-valued functions (logarithmic, complex power) and determine branches

- Find the Taylor series of a function and determine its circle or annulus of convergence;

- Compute the residue of a function and use the residue theory to evaluate a contour integral or an integral over the real line;

Attitude

- Evaluate a contour integral using parametrization, fundamental theorem of calculus and Cauchy’s integral formula