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Advanced Complex Analysis [1]

Paper Code: 
24MAT324(B)
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.
  2. Students also know about the infinite product of analytic functions, entire and meromorphic functions.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

 

24MAT

324(B)

 

 

 

Advanced Complex Analysis

 (Theory)

 

 

 

 

 

CO107: Develop analytical skills in solving problems related to conformal mapping, bilinear transformation.

CO108: Develop problem-solving skills by applying the Cauchy-Hadamard theorem to analyze and solve problems involving infinite series representations of functions.

CO109: Explore the Schwarz Lemma precisely and understand the related problems in complex analysis.

CO110: Gain proficiency in applying techniques of analytic continuation to study the behavior of holomorphic functions.

CO111: Explore related to infinite series and products, such as the theory of entire functions.

CO112: Contribute effectively in course- specific interaction.

 

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Mappings:
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: 
Power Series:
15.00

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: 
Entire function and related theorem:
15.00

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

Unit IV: 
Meromorphic function and analytic continuation:
15.00

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

 

Unit V: 
Infinite sums and infinite product of complex numbers:
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.

 

SUGGESTED READING

  • 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.

 

e- RESOURCES

 

JOURNALS

 

 

Academic Year: 
2024-2025 [7]

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Source URL: https://maths.iisuniv.ac.in/courses/subjects/advanced-complex-analysis-2

Links:
[1] https://maths.iisuniv.ac.in/courses/subjects/advanced-complex-analysis-2
[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_BwE
[6] https://www.springer.com/journal/11785
[7] https://maths.iisuniv.ac.in/academic-year/2024-2025