Advanced Complex Analysis

Paper Code: 
MAT324B
Objective: 

This course will enable the students to –

  1. To learn mapping properties of hypergeometric and some other special transcendental functions.
  2. Students also know about the infinite product of analytic functions, entire and meromorphic function.

 

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

MAT 324B

 

 

Advanced Complex Analysis

 (Theory)

 

 

 

 

 

The students will be able to –

 

CO105: Determine whether a sequence of analytic functions converges uniformly on compact sets.

CO106: Acquire knowledge about different types of functions viz. analytic, entire and meromorphic functions occur in complex analysis along with their properties

CO107: Describe conformal mappings between various plane regions.

CO108: Utilize the concepts of complex analysis to specific research problems in mathematics or other fields.

CO109: Enhance and develop the ability of using the language of mathematics in analyzing the real-world problems of sciences and engineering.

CO110: Express some functions as infinite series or products

 

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
I
15.00

Conformal mapping, bilinear transformation mappings, Special mappings: 

Unit II: 
II
15.00

Power Series: Absolute convergence, Cauchy’s Hadamard theorem, Circle and radius of convergence, Analyticity of the sum function or a power series, Complex inversion formula for inverse Laplace transform and related problems.

Unit III: 
III
15.00

Schwarz’s lemma and its consequences, Doubly periodic entire functions, Fundamental theorem of algebra, Zeros of certain polynomials.

Unit IV: 
IV
15.00

Meromorphic functions, Essential singularities and Picard’s theorem, Analytic continuation, Monodrmy theorem, Poisson integral formula, Analytic continuation via reflexion.

Unit V: 
V
15.00
Infinite sums and infinite product of complex numbers, Infinite product of analytic functions, Factorization of entire function.                                               
 
Essential Readings: 
  • S. Ponnusamy, Foundation of Complex Analysis, Narosa Publishing House, 2011.
  • L. R. Ahlofrs, Complex Analysis, Mc-Graw Hill,2017.
  • A.S.B. Holland, Introduction to the Theory of Entire Functions, Academic Press, 1973.
  • H.S. Kasana, Complex Variables: Theory and Applications, Prentice-Hall, New Delhi, 2005.
References: 
  • Mark J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press South Asian Edition, 2003.
  • J.W. Brown and R.V. Churchil, Complex Variables and Applications, McGraw Hill, New York, 2013.
  • R. Murray Spiegel, Theory and Problems of Complex variables, Schaum Outline Series, 2009.
  • K.K. Dubey, Fundamentals of Complex Analysis Theory and Applications, International Publishing House, 2009.
  • Joseph Bak and Donald J. Newman, Complex Analysis, Springer, 2010.
Academic Year: 
2023-2024
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