To introduce the student to a surprising, very beautiful theory having intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.

After completing this module students should:

 -  appreciate the central role of complex numbers in mathematics;

-  be familiar with all the classical holomorphic functions;

-  be able to compute Taylor and Laurent series of such functions;

-  understand the content and relevance of the various Cauchy formulae and theorems;

-  be familiar with the reduction of real definite integrals to contour integrals;

-  be competent at computing contour integrals.

Reminder of complex arithmetic and algebra, Holomorphicity, power series, radius of convergence, Elementary functions, solving basic equations, Contour integration and Cauchy theorem, Taylor and Laurent series, Poles and essential isolated singularities, The Residue Theorem, Evaluation of real integrals by means of contour integration.