The theory of complex functions is a very rich and classical part of applied analysis with important applications in many areas of physics, algebra and geometry. Considered will be: holomorphic (analytic) functions, line integrals, theorems of Cauchy and Liouville, Taylor and Laurent series, complex logarithm, residue calculus and its applications, Fourier and Laplace transformation and their ap- plications. For students interested in mathematical physics, algebra or number theory, spectral theory, control theory, differential geometry, random matrix theory, this course is crucial.