Course Details
Subject {L-T-P / C} : MA4103 : Complex Analysis - II { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Dr. Sangita Jha
Syllabus
Complex Functions, Spherical representations of extended complex plane, Analytic functions, Branches of multiple-valued functions, Complex Integral, Power Series, Laurent series and singularities, Residue calculus, Argument principle, Harmonic functions and the reflection principle, Conformal mappings, Geometry of Mobius transformations, Open mapping theorem, Maximum modulus theorem, Schwarz’s lemma, Partial fractions and factorization, Entire Functions, Picard Theorem.
Course Objectives
- To provide an overview of what is complex analysis and why it is important to study. To motivate how one can use the theory of complex analysis for evaluating many real analysis problems comfortably.
- To introduce analytic functions, complex integral, entire functions, conformal mappings, singularities, mapping theorems and their applications.
- To provide the fundamental concepts of complex analysis and <br />point out the differences between real and complex in each context.
- To motivate for higher studies in advance complex analysis.
Course Outcomes
After completing the course the students should be able to understand the basic theory of complex analysis and gain the knowledge to apply the fundamental results from complex analysis in modern mathematics and applied sciences. The students will have the knowledge and skills to solve problems independently. They will have the ability to use techniques from complex analysis and apply it to diverse filed like PDEs, Real Analysis, Fourier Transforms and Analytic Number Theory.
Essential Reading
- J. B. Conway, Functions of One Complex Variable, Springer , 1978
- T. W. Gamelin, Complex Analysis, Springer , 2000
Supplementary Reading
- J.E. Marsden and M.J. Hoffman, Basic Complex Analysis, W H Freeman & Co , 1998
- D.G. Zill and P.D. Shanahan, Complex Analysis, Jones and Bartlet Student Edition , 2003