The residue theorem and its applications

This text contains some notes to a three hour lecture in complex analysis given at Caltech. The lectures start from scratch and contain an essentially self-contained proof of the Jordan normal form theorem, I had learned from Eugene Trubowitz as an undergraduate at ETH Zürich in the third semester of the standard calculus education at that school. My text also includes two proofs of the fundamental theorem of algebra using complex analysis and examples, which examples showing how residue calculus can help to calculate some definite integrals. Except for the proof of the normal form theorem, the material is contained in standard text books on complex analysis. The notes assume familiarity with partial derivatives and line integrals. I use Trubowitz approach to use Greens theorem to prove Cauchy’s theorem. [ When I had been an undergraduate, such a direct multivariable link was not in my complex analysis text books (Ahlfors for example does not mention Greens theorem in his book).] For the Jordan form section, some linear algebra knowledge is required.