Faculty: Dr. Goutam Kumar Mahato

Course Name : Advanced Complex Analysis

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 Plan

Session 1

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

Session 2

Simply connected region
https://www.youtube.com/watch?v=hUw5RUeKasA

Session 3

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

Session 4

Residue
https://www.youtube.com/watch?v=EPhl1RlL78o
https://www.slideshare.net/Digvijaysinhgohil/cauchy-integral-theorem-amp-formula-complex-variable-numerical-method

Session 5

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

Session 6

Residue Theorem
https://www.youtube.com/watch?v=EPhl1RlL78o
https://www.slideshare.net/solohermelin/complex-variables-39353113

Session 7

The Argument Principle
https://www.youtube.com/watch?v=79-ESkh5_f0
https://www.slideshare.net/solohermelin/complex-variables-39353113

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 17

Harmonic Functions
https://www.youtube.com/watch?v=JQSC0lCPG24
https://fdocuments.in/document/complex6-harmonic-function-conjugate.html

Session 18

Conjugate Differential

Session 19

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

Session 20

Poisson's Formula
https://www.youtube.com/watch?v=3O0pw-l2MPI
https://www.slideshare.net/solohermelin/complex-variables-39353113

Session 21 & 22

Project-3 : Study on Harmonic Functions

Session 23

Taylor Series
https://www.youtube.com/watch?v=-55QChzD92w&t=187s
https://www.slideshare.net/littlepari351/taylor-series-29411223

Session 24

Taylor's Theorem
https://www.youtube.com/watch?v=-55QChzD92w&t=187s
https://www.slideshare.net/kath091583/taylors-theorem#:~:text=Taylor's%20Theorem%20gives%20an%20approximation,th%20derivative%20whenever%20it%20exists.

Session 25

Laurent Series
https://www.youtube.com/watch?v=0ZOMkmy-aTo
https://www.slideshare.net/aakash7497/laurents-series-types-of-singularities

Session 26

Laurent's Theorem
https://www.youtube.com/watch?v=RC15R-ktnUI
https://slideplayer.com/slide/5265544/

Session 27

Infinite Products
https://www.youtube.com/watch?v=xquEB6Q8yEk
https://www.slideshare.net/solohermelin/complex-variables-39353113

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 32

Riemann Zeta function
https://www.youtube.com/watch?v=sD0NjbwqlYw
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

Session 36

Simply periodic function
https://www.youtube.com/watch?v=Ns03ndh5Zzk
https://www.slideshare.net/UmangGupta5/fourier-series-31466525

Session 37

Module
https://www.youtube.com/watch?v=b4sfzHJeHsI

Session 38

Discrete module

Session 39

Unimodular Transformation

Session 40

Canonical basis

Session 41

Theorem on Canonical basis

Session 42 & 43

Project-6 : Study on Discrete modules