Advanced Complex Analysis

Teacher

Dr. Manju K Menon

Category

Department of Mathematics

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₹1,000.00

13 students
0 lessons
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10 week duration

13 students

Advanced Complex Analysis

Course Code: ME010301 Coordinator: Dr. Manju K. Menon

Course Credit: 4

Semester: III Class: MSc Mathematics

Course Type: Core

Department: Mathematics

COURSE OBJECTIVES

To make students learn the theory of Complex Analysis in continuation to what they have studies in their second semester Complex Analysis.

Learning Outcomes

After completing this course the learner should be able to

Definitions and Basic Properties of Harmonic Functions
Have a clear picture of the Mean-Value Property, Poisson’s Formula, Schwarz’s Theorem, The Reflection Principle
Have an idea about the Dirichlet’s Problem
Have a clear cut idea of Power Series Expansions – Weierstrass’s theorem, The Taylor Series, The Laurent Series
Definition and problems of Partial Fractions
Understand Infinite Products, Canonical Products
Derivation of the Gamma Function
Proofs of Jensen’s Formula, Hadamard’s Theorem
Understand the Riemann Zeta Function and its extension of to the Whole Plane
Derivation of the Functional Equation
Understanding the Normal Families
Proof of the Riemann Mapping Theorem
The Weierstrass’s P – function, zeta functions and sigma function and the derivation of Differential Equation.

Syllubus

Module 1: Harmonic Functions – Definitions and Basic Properties, The Mean-Value

Property, Poisson’s Formula, Schwarz’s Theorem, The Reflection Principle.

A closer look at Harmonic Functions – Functions with Mean Value Property,

Harnack’s Principle. The Dirichlet’s Problem – Subharmonic Functions, Solution of Dirichlet’s Problem ( Proof of Dirichlet’s Problem and Proofs of Lemma 1 and 2

excluded )

(Chapter 4 : Section 6: 6.1 – 6.5, Chapter 6 : Section 3 : 3.1 – 3.2 , Section

4 : 4.1 – 4.2)

Module 2: Power Series Expansions – Weierstrass’s theorem, The Taylor Series, The

Laurent Series Partial Fractions and Factorization – Partial Fractions, Infinite

Products, Canonical Products, The Gamma Function. Entire Functions –

Jensen’s Formula, Hadamard’s Theorem ( Hadamard’s theorem – proof

excluded)

(Chapter 5 : Section 1 : 1.1 – 1.3, Section 2 : 2.1 – 2.4, Section 3 : 3.1 – 3.2 )

Module 3: The Riemann Zeta Function – The Product Development, The Extension of

zeta(s) to the Whole Plane, The Functional Equation, The Zeroes of the Zeta

Function Normal Families – Normality and Compactness, Arzela’s Theorem

(Chapter 5 : Section 4 : 4.1 – 4.4, Section 5 : 5.2 – 5.3)

Module 4: The Riemann Mapping Theorem – Statement and Proof, Boundary

Behaviour, Use of the Reflection Principle, The Weierstrass’s Theory – The Weierstrass’s P- function and the other functions, The Differential Equation

(Chapter 6 : Section 1: 1.1-1.3, Chapter 7 : Section 3 : 3.1 – 3.3)

COURSE OUTLINE

No	Topics	Learning Outcomes	Hours
1	Harmonic Functions – Definitions and Basic Properties The Mean-Value Property Poisson’s Formula Schwarz’s Theorem The Reflection Principle. A closer look at Harmonic Functions – Functions with Mean Value Property Harnack’s Principle. The Dirichlet’s Problem – Subharmonic Functions Solution of Dirichlet’s Problem	· Understands the concept of Harmonic Functions and some major properties related to it. · Learns Harnack’s Principle. · Have an idea about Dirichlet’s Problem · Have an idea about harmonic functions	10 2 2 3
Seminar will be given to 1/4 th of students
2	Power Series Expansions – Weierstrass’s theorem The Taylor Series The Laurent Series Partial Fractions Infinite Products Canonical Products The Gamma Function Jensen’s Formula, Hadamard’s Theorem	· Have n idea about Power series · Understands Taylor’s and Laurent’s series · Have an idea about Infinite products ande studies theorems related to it. · Derives Gamma Function · Studies Jensen’s formula · Have an idea about Hadamard’s theorem	2 3 3 3 2 3
Seminar will be given to ¼ th of students
Internal I
3	The Riemann Zeta Function – The Product Development The Extension of zeta(s) to the Whole Plane The Functional Equation The Zeroes of the Zeta Function Normality and Compactness Arzela’s Theorem	· Will have a good idea about The Riemann Zeta Function · Understands zeta function and its properties · Concepts of Normality and Equicontinuity · Arzela’s Tgheorem	4 5 4 1
Seminar will be given to ¼ th of students students
4	The Riemann Mapping Theorem Boundary Behaviour -Use of the Reflection Principle The Weierstrass’s Theory – The Weierstrass’s P- function and the other functions The Differential Equation	· Studies The Riemann Mapping Theorem · Understands Boundary Behaviour · Studies Different types of functions and its properties · Derives Differential Equation	3 2 6 2
Seminar will be given to ¼ th of students students
Internal II

TEACHING SCHEDULE

TOPICS	HOURS	DATES	DESCRIPTION
Module I
Harmonic Functions – Definitions and Basic Properties	2	June 3, 4
The Mean-Value Property	2	June 5, 8
Poisson’s Formula	1	June 9
Schwarz’s Theorem	1	June 10
The Reflection Principle.	2	June 11, 14
A closer look at Harmonic Functions – Functions with Mean Value Property	2	June 15, 16
Harnack’s Principle.	3	June 17, 18, 19
The Dirichlet’s Problem – Subharmonic Functions	2	June 22, 23
Solution of Dirichlet’s Problem	2	June 24, 25
Seminars, Additional Discussion		Will Be Completed in June
Module II
Power Series Expansions – Weierstrass’s theorem	2	July 1, 2
The Taylor Series	2	July 3,7
The Laurent Series	1	July 8
Partial Fractions	1	July 9
Infinite Products	2	July 13, 14
Canonical Products	2	July 15, 16
The Gamma Function	3	July 20, 21, 22
Jensen’s Formula,	2	July 23, 24
Hadamard’s theorem	1	July 27
Seminars, Additional Discussion		Will Be Completed in July
Module III
The Riemann Zeta Function – The Product Development	2	August 4, 5
The Extension of zeta(s) to the Whole Plane	3	August 6, 7, 8
The Functional Equation	3	August 10, 11, 12
The Zeroes of the Zeta Function	2	August 18, 19
Normality and Compactness	3	August 20, 21, 24
Arzela’s Theorem	1	August 25
Seminars, Additional Discussion		Will Be Completed in July
Module IV
The Riemann Mapping Theorem	3	September 14. 15, 16
Boundary Behaviour -Use of the Reflection Principle	2	September 17,18
The Weierstrass’s Theory – The Weierstrass’s P- function and the other functions	5	September 21-25
The Differential Equation	3	September 28-30

PEDAGOGY

After the course the students will understand the advanced theories in Complex Analysis. The tutorials will focus on the understanding of concepts.

EVALUATION STRATEGY

The internal evaluation is based on the students attendance, seminar, Internal Assessments and assignment marks.

EVALUATION SCHEME

Sl. No	Components	Weightage
1	END TERM EXAM	80
2	ATTENDANCE	5
3	INTERNAL MARKS	10
4	SEMINARS/ ASSIGNMENT/ VIVA	5

TEXT BOOK :

Complex Analysis – Lars V. Ahlfors ( Third Edition ), McGraw Hill Book Company

REFERENCE BOOKS:

Chaudhary B., The Elements of Complex Analysis, Wiley Eastern.
Cartan H., Elementary theory of Analytic Functions of one or several variable, Addison

Wesley, 1973.

Conway J. B., Functions of one complex variable, Narosa publishing.
Lang S., Complex Analysis, Springer.
H. A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford, 1990.
Ponnuswamy S., Silverman H., Complex Variables with Application

WEB REFERENCE:

FACULTY DETAILS

Website: stpauls.ac.in

Email: [email protected]

Mobile: 9846335837

SEMINARS/ASSIGNMENTS

Topics will be announced in the online/offline class during the course.

Module 1

Harmonic Functions – Definitions and Basic Properties, The Mean-Value Property, Poisson’s Formula, Schwarz’s Theorem, The Reflection Principle. A closer look at Harmonic Functions – Functions with Mean Value Property, Harnack’s Principle. The Dirichlet’s Problem – Subharmonic Functions, Solution of Dirichlet’s Problem

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Module II

Power Series Expansions – Weierstrass’s theorem, The Taylor Series, The Laurent Series Partial Fractions and Factorization – Partial Fractions, Infinite Products, Canonical Products, The Gamma Function. Entire Functions – Jensen’s Formula, Hadamard’s Theorem

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No items in this section
Module III

The Riemann Zeta Function – The Product Development, The Extension of z (s) to the Whole Plane, The Functional Equation, The Zeroes of the Zeta Function Normal Families – Normality and Compactness, Arzela’s Theorem

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Module IV

The Riemann Mapping Theorem – Statement and Proof, Boundary Behaviour, Use of the Reflection Principle The Weierstrass’s Theory – The Weierstrass’s zeta function, The Differential Equation

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No items in this section

Dr. Manju K Menon

Dr. Manju K. Menon joined St. Paul's College, Kalamassery in 2005. She was an NBHM Research Fellow in 2002-2005 and completed her Ph D in Graph Theory under the guidance of Prof. A. Vijayakumar, CUSAT in 2010. She has 10 publications to her credit and successfully completed a Minor Research Project Funded by UGC. She was the convener of two UGC Sponsored National Seminars and a Compact Course organised in association with the Ramanujan Mathematical Society. She is currently the Controller of Examinations of St. Paul's College, Kalamassery. She is a life member of RMS, IMS and KMA. She is also a resource person in many National and International Seminars including the Third Indo Taiwan International Conference on Discrete Mathematics held at Taiwan in 2013. She is a research guide in Mathematics in Maharajas College, Ernakulam affiliated to Mahatma Gandhi University, Kottayam.

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