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Complex Analysis [1]

Paper Code: 
24DMAT601(A)
Credits: 
4
Contact Hours: 
4.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. 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.  
  2. Identify and construct complex-differentiable functions.

 

Course Outcomes: 

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

24DMAT

601(A)

Complex Analysis

(Theory)

 

 

 

 

CO67: Explain the concepts of complex variables with differentiability, Analyticity with Cauchy-Riemann equations.

CO68: Demonstrate techniques for integrating complex functions along curves in the complex plane by Cauchy’s integral theorem.

CO69:  Explore techniques of complex integration by integration formula and investigate power series with convergence properties.

CO70: Identify concept of singularities, Zeros, Residues and their application including evaluating complex integrals.

CO71: Compute definite integrals by using the residue theorem.

CO72: 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, Assigned tasks

 

 

Quiz, Class test, Individual and group task, Continuous assessment, Semester End Examination

 

 

Unit I: 
Complex plane, extended complex plane:
12.00

Stereographic projection, Complex valued functions Limit, continuity and differentiability, Analytic functions, C-R equations, Harmonic function, Construction of an analytic function.

 

 

Unit II: 
Complex integration and related theorems:
12.00

  Complex integration, Complex line integrals, Cauchy’s integral theorem, Cauchy’s fundamental theorem, Indefinite integrals, Fundamental theorem of integral calculus for complex function.

 

Unit III: 
Cauchy’s integral formula and theorems:
12.00

 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.

 

Unit IV: 
Concept of Singularity and Residues:
12.00

 Singularities, Meromorphic functions and entire functions, Riemann’s theorem, Casorati-Weierstrass theorem, Rouche’s theorem, Fundamental theorem of algebra, Residue at a singularity, Cauchy’s residue theorem.

 

Unit V: 
Contour integration:
12.00

Evaluation of real definite integral by contour integration (problems only).

 

Essential Readings: 
  • 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.

 

References: 
  • 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

 

JOURNALS 

Academic Year: 
2024-2025 [8]

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Source URL: https://maths.iisuniv.ac.in/courses/subjects/complex-analysis-12

Links:
[1] https://maths.iisuniv.ac.in/courses/subjects/complex-analysis-12
[2] https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf
[3] https://nptel.ac.in/courses/111103070
[4] https://nptel.ac.in/courses/111107056
[5] 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
[6] https://www.springer.com/journal/11785 
[7] https://link.springer.com/journal/40627 
[8] https://maths.iisuniv.ac.in/academic-year/2024-2025