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 which will allow students to work effectively with the concepts.
- Identify and construct complex-differentiable functions.
Course Outcomes (COs):
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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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DMAT601A |
Complex Analysis(Theory)
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The students will be able to – CO55: Explain the fundamental concepts of complex analysis and their role in modern mathematics and applied contexts. CO56: Demonstrate accurate and efficient use of complex analysis techniques. CO57: Apply the methods of complex analysis to evaluate definite integrals and infinite series CO58: Identify concept of singularities and zeros and apply these concept to find value of definite integral. CO59: Use the knowledge of mapping properties of elementary functions and discuss standard procedures on how to construct a mapping from one planar domain onto another. CO60: Apply problem-solving using complex analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts. |
Approach in teaching: Interactive Lectures, Discussion, Power Point Presentations, Informative videos
Learning activities for the students: Self learning assignments, Effective questions, presentations, Giving tasks |
Quiz, Poster Presentations, Power Point Presentations, Individual and group projects, Open Book Test, Semester End Examination
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Complex plane, Extended complex plane: 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.
