Advanced Complex Analysis

Teacher

Dr. Goutam Kumar Mahato

Category

Core Courses

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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.

Course Outcomes

COs Course outcomes Mapping Cos with POs (High-3, Medium-2, Low-1)
CO1 Get a deep understanding of the fundamental concepts of Residues, Laurent series, Harmonic and Periodic functions.

 

PO1(3), PO2(2), PO9(3)
CO2 Evaluate definite and improper real integrals applying the Cauchy's Residue Theorem.

 

PO1(3),

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  

Our Main Teachers

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 […]