Complex Analysis

This is a first course in Complex Analysis focussing on holomorphic functions and its basic properties like Cauchy’s theorem and residue theorems, the classification of singularities, and the maximum principle. We shall study the singularities of holomorphic functions. If time permits, we shall also study Branches of the complex logarithm through covering spaces and attempt proving Picard’s theorem. INTENDED AUDIENCE :Third year Undergraduate or first year Master’s students in various universities.PREREQUISITES :Real Analysis, Linear AlgebraINDUSTRIES SUPPORT :Almost all engineering-based companies

Week 1: Algebra and Topology of the complex plane
Week 2:Geometry of the complex plane, Complex differentiation
Week 3: Power series and its convergence, Cauchy-Riemann equations
Week 4: Harmonic functions,Möbius transformations
Week 5:Integration along a contour, The fundamental theorem of calculus
Week 6: Homotopy, Cauchy’s theorem
Week 7: Cauchy integral formula, Cauchy’s inequalities and other consequences
Week 8: Winding number, Open mapping theorem, Schwarz reflection Principle
Week 9: Singularities of a holomorphic function, Laurent series
Week 10: The residue theorem, Argument principle, Rouche’s theorem
Week 11: Branch of the Complex logarithm, Automorphisms of the Unit disk
Week 12: Covering spaces, Picard's theorem