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Advanced Complex Analysis (Optional Paper) [1]

Paper Code: 
MAT 427
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Unit I: 
I
15.00
Schwarz’s Lemma and its consequences, Doubly periodic entire functions, Fundamental theorem of Algebra, Zeros of certain polynomials.                                                        (15 Lectures) 
 
Unit II: 
II
15.00

Meromorphic functions, Essential Singularities and Picard’s Theorem, Analytic Continuation, Monodrmy theorem, Poisson integral formula, Analytic continuation via reflexion. 

Unit III: 
III
15.00
Infinite sums and infinite product of complex numbers, Infinite product of analytic functions, Factorization of entire functions.                                                                (15 Lectures) 
 
Unit IV: 
IV
15.00

The Gamma functions, The Zeta functions, Jensen Formula, The order and the Genus of entire functions. Open mapping theorem and Hurwitz’ theorem, Basic results on univalent functions.  

Unit V: 
V
15.00
 Normal families, The Riemann mapping theorem, Biberbach Conjecture, The Bloch-Landau Theorem, Picard’s Theorem Open mapping theorem and Hurwitz’ theorem, Basic results on univalent functions.                                                                                                       
 
Essential Readings: 
  1. S. Ponnusamy, Foundation of Complex Analysis, 2nd edition, Narosa Publishing House. 
  2. L. R. Ahlofrs, Complex Analysis, McGraw Hill 
  3. A. S. B. Holland, Introduction to the theory of entire functions, Academic Press. 
 
Academic Year: 
2014-2015 [2]

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[1] https://maths.iisuniv.ac.in/courses/subjects/advanced-complex-analysis-optional-paper-4
[2] https://maths.iisuniv.ac.in/academic-year/2014-2015