Course Objectives

To understand the applications of Residue for evaluation of definite and improper integrals occurring in Real analysis and Applied mathematics.

To know about special functions like Riemann zeta function which plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

Course Outcomes

Course Syllabus

MODULE – I (3hr+0hr+0hr)

Index of a point with respect to a closed curve, Simply connected region, General statement of Cauchy's theorem.

MODULE – II (4hr+0hr+2hr)

Residue, process for finding out the residues, Residue theorem, the Argument Principle.

PROJECT 1: Study on Residues and their applications.

MODULE – III (5hr+0hr+2hr)

Definite Integrals: Evaluation of definite integrals (Types - 1, 2, 3, 4, 5).

PROJECT 2: Evaluation of different types of real definite integrals using Residue theorem.

MODULE – IV (4hr+0hr+2hr)

Harmonic functions, conjugate differential, The Mean-Value Property, Poisson's formula.

PROJECT 3: A study on Harmonic functions.

MODULE – V (6hr+0hr+2hr)

Taylor Series, Taylor's theorem, Laurent series, Laurent's theorem, infinite products, theorems on infinite products.

PROJECT 4: A study on Laurent series expansion of different types of meromerphic functions.

MODULE – VI (3hr+0hr+2hr)

Entire functions: Jensen's formula, Riemann Zeta function, theorem on Riemann Zeta function.

PROJECT 5: A study on Riemann Zeta function and it's properties.

MODULE – VII (6hr+0hr+2hr)

Simply periodic function, Module, Discrete module, Unimodular transformation, Canonical basis, theorem on Canonical basis.

PROJECT 6: A study on discrete modules.

 BOOK PRESCRIBED

1 L. V. Ahlfors, “Complex Analysis”, McGraw-Hill, Inc.

Chapters: 4 (2.1, 4.2 to 4.5, 5.1 to 5.3, 6.1 to 6.3), 5 (1.2, 1.3, 2.2, 3.1, 4.1), 7 (1.1, 2.1, 2.2, 2.3)

Session Plan