https://studiegids.vu.nl/en/courses/2024-2025/X_400386The student can decide whether a complex function is analytic (=differentiable in the complex sense) and knows the connection with the Cauchy-Riemann equations.They can do computations with elementary functions such as exp/log/sin/cos over the complex numberts.They can integrate analytic functions along a path on the complex plane, using the theorem of Cauchy-Goursat and its corollaries.They can compute Laurent series and determine the type of singularities of analytic functions.They can compute integrals of complex functions using the residue theorem and know how to use this to compute integrals of real functions.In complex analysis one generalizes the standard concepts of real analysis such as differentiation and integration from the real line to the complex plane. Although these generalizations arise very naturally and all standard examples of functions are also differentiable in the complex sense, the latter property surprisingly turns out to be much stronger. As a consequence, complex differentiable functions immediately obey very special properties which we are going to explore in this course. In particular, they lead to completely new and efficient methods for computing integrals of real functions. During the lectures the following topics will be treated:complex differentiation and the Cauchy-Riemann equationscomplex integration and the theorem of Cauchy-Goursatelementary properties of complex differentiable functionssingularities, Laurent series and the residue theoremapplication to integrals of real functionsLecture (2 hours) and tutorial class (2 hours)Two written exams (40%+40%) and two hand-in homeworks (10%+10%). The retake exam counts for 100% of the final grade.Churchill, R. V., & Brown, J. W.: Complex variables and applications. Ninth edition, 2014, McGraw-Hill Book Co., New YorkBachelor Mathematics Year 2Calculus, Analysis, Linear algebra are necessary background.