AIS

ATML in Complex Analysis, 2007

Proposed Syllabus and Speakers

Resource Persons:

Speakers

Prof. K. D. Joshi kdj at math.iitb.ac.in

Prof. S. A. Katre sakatre at math.unipune.ernet.in

Dr. H. Bhate hbhate at math.unipune.ernet.in

Prof. A. R. Shastri ars at math.iitb.ac.in
Invited Speakers

V. D. Sharma, IIT Bombay: Application of Complex Analysis in Fluid Dynamics

Gopal Srinivasan, IIT Bombay: Gamma function, Riemann zeta function

Mangala Narlikar, BIM: Gaussian Sums

Course Associates

1. Sameer Chavan, IISER, Pune.	sameer_chavan95 at yahoo.co.in
2. Pratul Gadagkar, Univ. of Pune, Pune	pratul1980 at math.unipune.ernet.in
3. Vinay Wagh, IIT, Bombay, Mumbai	vinay at math.iitb.ac.in

Syllabus

Basic algebraic and geometric properties of complex numbers, rigid motions. Open sets, Connectivity and path connectivity.
Complex differentiability, CR-equations, holomorphic anti-holomorphic functions, operators $\frac{\partial}{\partial z},\frac{\partial}{\partial \bar{z}},$ harmonic functions.
Review of sequences and series, Uniform convergence, Weierstrass' M-test, power series, Analytic functions, exponential and trigonometric functions logarithmic function, complex exponents of complex numbers.
Contour integration, existence of primitives, Cauchy-Goursat theorem, winding number, Cauchy's theorem, Cauchy's integral formulae, Morera's theorem.
Local behaviour of Analytic functions, Taylor series. zeros of analytic functons, identity theorem, Analysis of isolated singularities. Poles and residues. Laurent Expansion, Casorati Weierstrass' theorem. Open mapping theorem. Maximum modulus principle. Schwarz' lemma.
Runge's Approximation Theorem, Generalization of Cauchy's theorem, Argument principle, simple connectivity, Residue theorem, Integral formula for Inverse function, Holomorphicity of Symmetric functions.
Computation of real integrals.
Convergene in function Theory, Hurwitz theorem, Mittag-Leffler theorem, partial fraction development, infinite products, Weierstrass's theorem, Gamma-function and Riemann zeta-function.
Normal families, Equicontinuity, Arzela-Ascoli, Montels theorem. Riemann mapping theorem.
Harmonic functions, Poisson Integral, Schwarz's reflection principle, Harnack' principle, Sub-harmonicity, Dirichlet's problem, Greens's function, another proof of Riemann mapping theorem.

References:

Ahlfors, V., Complex Analysis MacGraw-Hill,

Remmert, R., Theory of Complex Functions, GTM-122 Springer.

Rudin, W., Real And Complex Analysis, MacGraw-Hill 2nd ed.

Shastri A. R., An introduction to Complex Analysis, MacMillan India, 1999.

Prof. K. D. Joshi	kdj at math.iitb.ac.in
Prof. S. A. Katre	sakatre at math.unipune.ernet.in
Dr. H. Bhate	hbhate at math.unipune.ernet.in
Prof. A. R. Shastri	ars at math.iitb.ac.in