Course No
M308
Credit
4
Syllabus
Algebraic and geometric representation of complex numbers; elementary functions including the exponential functions and its relatives (log, cos,sin, cosh, sinh, etc.); concept of holomorphic (analytic) functions, complex derivative and the Cauchy-Riemann equations; harmonic functions. Conformal Mapping, Linear Fractional Transformations, Complex line integrals and Cauchy Integral formula, Representation of holomorpic functions in terms of power series, Morera’s theorem, Cauchy estimates and Liouville’s theorem, zeros of holomorphic functions, Uniform limits of holomorphic functions. Behaviour of holomorphic function near an isolated singularity, Laurent expansions, Counting zeros and poles, Argument principle, Rouche’s theorem, Calculus of residues and evaluation of integrals using contour integration. The Open Mapping theorem, Maximum Modulus Principle, Schwarz Lemma.
Text Books
- J. B. Conway, “Functions of One Complex Variable”, Narosa Publishing House, 2002.
- R. E. Greene, S. G. Krantz, “Function Theory of One Complex Variable”, American Mathematical Society, 2011.
Reference Books
- W. Rudin, “Real and Complex Analysis”, Tata McGraw-Hill, 2013.
- L. V. Ahlfors, “Complex Analysis”, Tata McGraw-Hill, 2013.
- T. W. Gamelin, “Complex Analysis”, Undergraduate Texts in Mathematics, Springer, 2006.
- E. M. Stein, R. Shakarchi, “Complex Analysis”, Princeton University Press, 2003.