Complex Analysis

Paper Code: 
25DMAT601(A)
Credits: 
4
Contact Hours: 
60.00
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

25DMAT

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
 
 
Academic Year: 
2025-2026
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