Curriculum and Syllabi - Indian Institute of Technology Bhubaneswar

9. Complex Analysis (MA4007):3-0-0: 3 Credits Prerequisite: RealAnalysisPolar representation and roots of complex numbers; Spherical representation of extended complexplane; Elementary properties and examples of analytic functions: The exponential, Trigonometricfunctions, Mobius transformations, Cross ratio; Complex integration: Power series representation ofanalytic functions, Zeros of analytic functions, Cauchy theorem and integral formula, The index of apoint with respect to a closed curve, the general form of Cauchy’s theorem; Open Mapping Theorem;Classification of singularities: Residue theorem and applications; The Argument Principle; TheMaximum modulus Principle; Schwarz’s lemma; Phragmen-Lindelof theorem.Texts:1. J.B. Conway, Functions of One Complex Variable, 2nd ed., Narosa, New Delhi, 1978.2. L. V. Ahlfors, Complex Analysis, 3rd edition, McGraw Hill, 1979.References3. T.W. Gamelin, Complex Analysis, Springer International Edition, 2001.4. R.V. Churchill and J.W. Brown, Complex Variables and Applications, 5th edition, McGraw Hill, 1990.5. W. Rudin, Real and complex analysis. McGraw-Hill Book Co., 1987.10. Topology (MA4008):3-1-0: 4 Credits Prerequisite: RealAnalysisTopological spaces, Basis and subbasis, The order topology, Subspace topology, Closed sets.Countability axioms, Limit points, Convergence of nets in topological spaces, Continuous functions,homomorphisms. The product topology, box topology, Metric topology, Quotient topology.Connected spaces, Connected sets in R, Components and path components, Compact spaces,Compactness in metric spaces, Local compactness, One point compactification. Separation axioms,Uryshon’s lemma, Uryshon’s metrization theorem, Tietz extension theorem. The Tychonoff theorem,Completely regular spaces, Stone -Czech compactification.Texts:1.J.R. Munkres, Topology, 2nd Ed., Pearson Education (India), 2001.2. H. L. Royden, Real Analysis, 3rd edition, Prentice Hall of India, 1995.References:1. M. A. Armstrong, Basic Topology, Springer(India), 2004.2. J.L. Kelley, General Topology, Van Nostrand, Princeton, 1955.3. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, New York,1963.11. Ordinary Differential Equations (MA4009):3-0-0: 3 Credits Prerequisite: NilOrdinary differential equations- first order equations, Picard’s theorem (existence and uniqueness ofsolution to first order ordinary differential equation). Second order differential equations- second orderlinear differential equations with constant coefficients. Systems of first order differential equations,equations with regular singular points, stability of linear systems. Introduction to power series andpower series solutions. Special ordinary differential equations arising in physics and some specialfunctions (e.g. Bessel’s functions, Legendre polynomials, Gamma functions) and their orthogonality.Oscillations - Sturm Liouville theory.