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 applications. For students interested in mathematical physics, algebra or number theory, spectral theory, control theory, differential geometry, image analysis and digital signal processing, this course is crucial.