Course Archives Theoretical Statistics and Mathematics Unit | ||||||||
Course: Complex Analysis Instructor: Sury B Room: G25 Level: Postgraduate Time: Currently offered |
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Syllabus Past Exams Syllabus: i) Review of sequences and series of functions including power series, Complex differentiation and Cauchy-Riemann equation, Cauchys theorem and Cauchys integral formula, Power series expansion of holomorphic function, zeroes of holomorphic functions, Maximum Modulus Principle, Liouvilles Theorem, Moreras Theorem. ii) Complex logarithm and winding number, Singularities, Meromorphic functions, Casorati Weierstrass theorem, Riemann sphere, Laurent series, Residue Theorem and applications to evaluation of definite integrals, Open Mapping Theorem, Rouches Theorem. iii) Conformal maps, Schwarz lemma, Linear fractional transformations, automorphisms of a disc, Introduction to Gamma function. iv) Equicontinuity and Arzela-Ascoli Theorem, Normal family, Montels theorem and Riemann mapping theorem. v) If time permits then the following topics can also be covered: Mittag-Leffler Theorem, Infinite product, Weierstrass factorization theorem. Suggested Texts : (a) Complex Analysis - L. Ahlfors. (b) Elementary Theory of Analytic Functions of One or Several Complex Variables - H. Cartan . (c) Complex Analysis - E. M. Stein, R. Shakarchi. (d) Complex Analysis - D. Sarason. Evaluation:
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Midterm
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