COMPLEX ANALYSIS
Complex plane, Extended complex plane: Stereographic projection, Complex valued functions (Limit, continuity and differentiability), Analytic functions, C-R equations, Harmonic function, Construction of an analytic function.
Complex integration, Complex line integrals, Cauchy’s integral theorem, Cauchy’s fundamental theorem, Indefinite integrals, Fundamental theorem of integral calculus for complex function.
Cauchy’s integral formula, Analyticity of the derivative of an analytic function, Liouville’s theorem, Poisson’s integral formula, Morera’s theorem, Maximum modulus principle. Taylor’s and Laurent’s series.
Singularities, Branch points, Meromorphic functions and entire functions, Riemann’s theorem, Casorati-Weirstrass theorem, Rouche’s theorem, Fundamental theorem of algebra, Residue at a singularity, Cauchy’s residue theorem.
Evaluation of real definite integral by contour integration (problems only).
- G. N. Purohit, S.P. Goyal, Complex Analysis, Jaipur Publishing House, 2015.
- H. S. Kasana, Complex Variables: Theory and Applications, Prentice Hall, Delhi, 2005.
- S. Ponnuswamy, Introduction to Complex Analysis, Narosa Publishers, 2011.
- P. K. Banerji, V. B. L. Chaurasia, S. P. Goyal, Functions of a Complex Variable, RBD Publishing House, 2017.
- R. Murray Spiegel, Theory and Problems of Complex Variables, Schaum Outline Series,1974.
- K.K. Dubey, Fundamentals of Complex Analysis Theory and Application, International Publishing House, 2009.
- Rolf Nevalinna, Veikko Paatero, Introduction to Complex Analysis, AMS Chelsea Publishing, 2007.
- Joseph Bak and Donald J. Newman, Complex Analysis, Springer, 2010.
- James Ward Brown, Ruel V. Churchill, Complex Variables and Application, McGraw Hills Book Co., 2010.
