Complex Function Theory is one of the classical mathematical courses. It is on the one hand an extension of real analysis, but on the other hand it has its own (surprising) results, which do not have a counterpart in real analyses. A central role in the course is played by the (contour)integral. Before we can introduce this, we need to lay a basis by introducing elementary functions, defining differentiability (analytic). The contour integral can be used to solve ``nasty’’ real integrals, but also leads to the residue theorem. This residue theorem can be used to show that every analytic function is infinitely many times differentiable, which leads to the Taylor series of an analytic function, and to the Laurent series of a meromorphic function. Furthermore, it is at the basis of the Principle of Argument, and the Rouché Theorem, which counts the number of zeros inside a domain. Finally, we show that the inverse Laplace transform can be found using this residue theorem.|
The course is presented in a combination of lectures and tutorials during the first part of block 2B.