Complex Plane, connected and compact sets. Extended complex plane: Stereographic projection. Complex valued functions, Analytic functions, C-R equation, Harmonic functions. Construction of an analytic functions.
Unit II:
II
9.00
Complex Integration, Complex line Integrals, Cauchy’s integral theorem, Cauchy’s Fundamental theorem, Indefinite integrals, fundamental theorem of Integral Calculus for complex functions.
Unit III:
III
9.00
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.
Unit IV:
IV
9.00
Singularities. Meromorphic functions and Entire functions. Riemann’s theorem, Casorati-Weirstrass theorem. Rouche’s theorem. Fundamental theorem of algebra. Cauchy’s Residue theorem.
Unit V:
V
9.00
Contour integration. Conformal mapping (Introduction).Bilinear transformation and their simple properties . Power series.
Essential Readings:
Complex Analysis , Purohit and Goyal , Jaipur Publishing House
Complex Variables: Theory and Applications ,H.S.Kasana, Prentice Hall, Delhi
Introduction to Complex Analysis, S.Ponnuswamy, Narosa Publishers.
Complex Variables and Application, Brown and Churchill, McGraw Hills Book Co.
References:
Theory and Problems of Complex Variables, R.Murray Spiegel , Schaum Outline Series
