ADVANCED COMPLEX ANALYSIS (Optional Paper)
- To learn mapping properties of hypergeometric and some other special transcendental functions. Students also know about infinite product of analytic functions, entire and meromorphic functions.
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Learning Outcomes |
Learning and teaching strategies |
Assessment |
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After the completion of the course the students will be able to: CLO80- Determine whether a sequence of analytic functions converges uniformly on compact sets. CLO81- Acquire knowledge about different types of functions viz. analytic, entire and meromorphic functions occur in complex analysis along with their properties CLO82- Describe conformal mappings between various plane regions. CLO83- Utilize the concepts of complex analysis to specific research problems in mathematics or other fields. CLO84- Enhance and develop the ability of using the language of mathematics in analyzing the real-world problems of sciences and engineering. CLO85- Express some functions as infinite series or products CLO86- Expand some simple functions as their Taylor and Laurent series, classify the nature of singularities, find residues and apply Cauchy Residue theorem to evaluate integrals. |
Approach in teaching: Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching Learning activities for the students: Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical |
Presentations by Individual Students. Class Tests at the end of each unit. Written assignment(s) Semester End Examination |
Conformal mapping, bilinear transformation mappings, Special mappings:
W(z)=1/z, z+1/z, Sin(z), Cos(z), Z2 , ez
Power Series: Absolute convergence, Cauchy’s Hadamard theorem, Circle and radius of convergence, Analyticity of the sum function or a power series, Complex inversion formula for inverse Laplace transform and related problems.
Schwarz’s lemma and its consequences, Doubly periodic entire functions, Fundamental theorem of algebra, Zeros of certain polynomials.
Meromorphic functions, Essential singularities and Picard’s theorem, Analytic continuation, Monodrmy theorem, Poisson integral formula, Analytic continuation via reflexion.
Infinite sums and infinite product of complex numbers, Infinite product of analytic functions, Factorization of entire function.
- S. Ponnusamy, Foundation of Complex Analysis, Narosa Publishing House, 2011.
- L. R. Ahlofrs, Complex Analysis, Mc-Graw Hill, 1979.
- A.S.B.Holland, Introduction to theTtheory of Entire Functions, Academic Press, 1973.
- H.S.Kasana, Complex Variables: Theory and Applications, Prentice-Hall, New Delhi, 2005.
- Mark J. Ablowitz, A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press South Asian Edition, 1998.
- J.W. Brown, R.V. Churchil, Complex Variables and Applications, McGraw Hill, New York, 1990.
- R. Murray Spiegel,Theory and Problems of Complex variables, Schaum Outline Series,1974.
- K.K. Dubey, Fundamentals of Complex Analysis Theory and Applications, International Publishing House, 2009.
- Joseph Bak, Donald J. Newman, Complex Analysis, Springer, 2010.
