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ADVANCED COMPLEX ANALYSIS (Optional Paper) [1]

Paper Code: 
MAT 324B
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. To learn mapping properties of hypergeometric and some other special transcendental functions. Students also know about infinite product of analytic functions, entire and meromorphic functions.

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Paper Code

Paper Title

MAT 324B

 

 

 

 

 

 

Advanced Complex Analysis

 (Theory)

 

 

 

 

 

The students will be able to –

 

CO86: Determine whether a sequence of analytic functions converges uniformly on compact sets.

CO87: Acquire knowledge about different types of functions viz. analytic, entire and meromorphic functions occur in complex analysis along with their properties

CO88: Describe conformal mappings between various plane regions.

CO89: Utilize the concepts of complex analysis to specific research problems in mathematics or other fields.

CO90: Enhance and develop the ability of using the language of mathematics in analyzing the real-world problems of sciences and engineering.

CO91: Express some functions as infinite series or products

CO92: Expand some simple functions as their Taylor and Laurent series, classify the nature of singularities, find residues and apply Cauchy Residue theorem to evaluate integrals.

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: W(z) =1/z ,z+1/z, z^2, e^z, Sin (z), Cos (z).
 
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, 1979.
  • 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.
  • Mark J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press South Asian Edition, 1998.
  • J.W. Brown and R.V. Churchil, Complex Variables and Applications, McGraw Hill, New York, 1990.
  • R. Murray Spiegel, Theory and Problems of Complex variables, Schaum Outline Series, 1974.
  • 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: 
2020-2021 [2]

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Source URL: https://maths.iisuniv.ac.in/courses/subjects/advanced-complex-analysis-optional-paper-3

Links:
[1] https://maths.iisuniv.ac.in/courses/subjects/advanced-complex-analysis-optional-paper-3
[2] https://maths.iisuniv.ac.in/academic-year/2020-2021