Complex Analysis
This course will enable the students to -
- Introduce the fundamental ideas of the functions of complex variables, developing a clear understanding of the fundamental concepts of complex analysis such as analytic functions, complex integrals and a range of skills that will allow students to work effectively with the concepts.
- Identify and construct complex-differentiable functions.
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Course |
Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Course Code |
Course Title |
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24CMAT 611 |
Complex Analysis(Theory)
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CO123: Explain the fundamental concepts of complex variables that are differentiable, focusing on the concept of analyticity and the Cauchy-Riemann equations. CO124: Demonstrate techniques for integrating complex functions along curves in the complex plane by Cauchy’s integral theorem. CO125: Explore techniques of complex integration by complex integration formula and investigate power series representations of analytic functions. CO126: Identify concept of singularities, zeros residues and their application, including evaluating complex integrals. CO127: Evaluate definite integrals by using the residue theorem. CO128: Contribute effectively in course-specific interaction. |
Approach in teaching: Interactive Lectures, Discussion, Power Point Presentations, Informative videos
Learning activities for the students: Self learning assignments, Effective questions, presentations, Assigned tasks |
Quiz, Individual and group projects, Open Book Test, Semester End Examination
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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, 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 and 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 and S. P. Goyal, Functions of a Complex Variable, RBD Publishing House, 2017.
- R. Murray Spiegel, Theory and Problems of Complex Variables, Schaum Outline Series, 2000.
- K. K. Dubey, Fundamentals of Complex Analysis Theory and Application, International Publishing House, 2009.
- Rolf Nevalinna and Veikko Paatero, Introduction to Complex Analysis, AMS Chelsea Publishing, 2007.
- Joseph Bak and Donald J. Newman, Complex Analysis, Springer, 2010.
- James Ward Brown and Ruel V. Churchill, Complex Variables and Application, McGraw Hills Book Co., 2010.
e- RESOURCES
- https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf
- https://nptel.ac.in/courses/111103070
- https://nptel.ac.in/courses/111107056
JOURNALS
- https://www.hindawi.com/journals/jca/utm_source=google&utm_medium=cpc&utm_campaign=HDW_MRKT_GBL_SUB_ADWO_PAI_DYNA_JOUR_JMATH_X0000&gclid=CjwKCAjwsJ6TBhAIEiwAfl4TWH6IKXbi9LgWWuhTsx4vLdrs-S2zjdS65-EQS7DOVhGN8fb1fJX18hoCcXYQAvD_Bw
- https://www.springer.com/journal/11785
- https://link.springer.com/journal/40627
