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# Code(Credit) : CUTM1535(3-0-1)

## 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.

## Learning Outcomes

Upon successful completion of this course, students will be able to:

Get a deep understanding of the fundamental concepts of Residues, Laurent series, Harmonic and Periodic functions.

Evaluate definite and improper real integrals applying the Cauchy's Residue Theorem.

## 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 1

Index of a point with respect to a closed curve https://www.youtube.com/watch?v=-8oX7abv_c4

## Session 3

General statement of Cauchy's theorem https://www.youtube.com/watch?v=-kKcruyhq3c

## Session 5

Process for finding out the residues https://www.youtube.com/watch?v=ZyOIZq__zxE

## Session 8 & 9

Project-1 : Study on Residues and their Applications

## Session 10

Evaluation of Definite Integrals (Type-1) https://www.youtube.com/watch?v=nfNzT2SpN4g https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 11

Evaluation of Definite Integrals (Type-2) https://www.youtube.com/watch?v=nn5Dd-1BXH4 https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 12

Evaluation of Definite Integrals (Type-3) https://www.youtube.com/watch?v=b8VMghnSVxI https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 13

Evaluation of Definite Integrals (Type-4) https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 14

Evaluation of Definite Integrals (Type-5) https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 15 & 16

Project-2 : Evaluation of different types of real definite integrals using Residue theorem

## Session 18

Conjugate Differential

## Session 19

The Mean-value Property https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 21 & 22

Project-3 : Study on Harmonic Functions

## Session 28

Theorems on Infinite Products https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 29 & 30

Project-4 : Study on Laurent series expansion of different types of meromerphic functions

## Session 31

Entire Functions: Jensen's Formula https://www.youtube.com/watch?v=4r0_mpjrdiI https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 33

Theorem on Riemann Zeta function https://www.youtube.com/watch?v=LFwSIdLSosI https://www.slideshare.net/solohermelin/complex-variables-39353113

## Session 34 & 35

Project-5 : Study on Riemann Zeta function and it's properties

Discrete module

## Session 39

Unimodular Transformation

Canonical basis

## Session 41

Theorem on Canonical basis

## Session 42 & 43

Project-6 : Study on Discrete modules

### Dr. Goutam Kumar Mahato

##### Teaching
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Received M.Sc. (Mathematics) from Vinoba Bhave University, Hazaribag and B.Ed. from Ranchi University, Ranchi in the year of 2006 and 2008 respectively. Awarded PhD, in the year of 2013, from Indian School of Mines, Dhanbad. Published 22 research papers in different journals of international repute. Participated and presented 16 research papers in various National and […]