Syllabus
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Course Code: MMATH21-405 Course Name: Elective IV) Advanced Complex Analysis |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Convex functions and Hadamard’s three circles theorem, Phragmen-Lindelöf theorem. Spaces of continuous functions, Arzela-Ascoli theorem, Spaces of analytic functions, Hurwitz’s theorem, Montel’s theorem, Spaces of meromorphic functions. | |
| 2 | Riemann mapping theorem, Weierstrass’ factorization theorem, Factorization of sine function, Gamma function and its properties, functional equation for gamma function, Bohr-Mollerup theorem, Reimann-zeta function, Riemann’s functional equation, Euler’s theorem. | |
| 3 | Runge’s theorem, Simply connected regions, Mittag-Leffler’s theorem. Analytic continuation, Power series method of analytic continuation , Schwarz reflection principle. Monodromy theorem and its consequences. Harmonic functions, Maximum and minimum principles, Harmonic function on a disk, Harnack’s theorem, Sub-harmonic and super-harmonic functions, Dirichlet’s problems, Green’s function. |
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| 4 | Entire functions :Jensen’s formula, Poisson–Jensen formula. The genus and order of an entire function, Hadamard’s factorization theorem. The range of an analytic function : Bloch’s theorem, Little-Picard theorem, Schottky’s theorem, Montel-Carathedory theorem, Great Picard theorem. |
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