Introduction : Introduction and overview of the course, lecture-wise description – The Algebra Geometry and Topology of the Complex Plane : Complex numbers, conjugation, modulus, argument and inequalities – Powers and roots of complex numbers, geometry in the complex plane, the extended complex plane – Topology of the complex plane: Open sets, closed sets, limit points, isolated points, interior points, boundary points, exterior points, compact sets, connected sets, sequences and series of complex numbers and convergence.
Complex Functions: Limits, Continuity and Differentiation : Introduction to complex functions – Limits and continuity – Differentiation and the Cauchy-Riemann equations, analytic functions, elementary functions and their mapping properties, harmonic functions – Complex logarithm multi-function, analytic branches of the logarithm multi-function, complex exponent multi-functions and their analytic branches, complex hyperbolic functions – Problem Session
Complex Integration Theory : Introducing curves, paths and contours, contour integrals and their properties, fundamental theorem of calculus – Cauchy’s theorem as a version of Green’s theorem, Cauchy-Goursat theorem for a rectangle, The anti-derivative theorem, Cauchy-Goursat theorem for a disc, the deformation theorem – Cauchy’s integral formula, Cauchy’s estimate, Liouville’s theorem, the fundamental theorem of algebra, higher derivatives of analytic functions, Morera’s theorem – Problem Session
Further Properties of Analytic Functions : Power series, their analyticity, Taylor’s theorem – Zeroes of analytic functions, Rouche’s theorem – Open mapping theorem, maximum modulus theorem.
Mobius Transformations : Properties of Mobius transformations – Problem Session
Isolated Singularities and Residue Theorem : Isolated singularities, removable singularities – Poles, classification of isolated singularities – Casoratti-Weierstrass theorem, Laurent’s theorem – Residue theorem, the argument principle – Problem Session
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