Complex Analysis

Paper Code: 
CMAT 611
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 (COs):

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

CMAT611

 

 

 

 

 

 

Complex Analysis

(Theory)

 

 

 

 

The students will be able to –

 

CO86: Explain the fundamental concepts of complex analysis and their role in modern mathematics and applied contexts.

CO87: Demonstrate accurate and efficient use of complex analysis techniques.

CO88: Apply the methods of complex analysis to evaluate definite integrals and infinite series

CO89: Identify concept of singularities and zeros and apply these concept to find value of definite integral.

CO90: 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.

CO91: 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

 

 

Unit I: 
I
9.00
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.
 
Unit II: 
II
9.00

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: 
III
9.00

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: 
IV
9.00

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.

Unit V: 
V
9.00

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.
Academic Year: 
2023-2024
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