Advanced Complex Analysis

Dr. Manju K Menon
Mathematics
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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:

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

Wesley, 1973.

  1. Conway J. B., Functions of one complex variable, Narosa publishing.
  2. Lang S., Complex Analysis, Springer.
  3. H. A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford, 1990.
  4. 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

    No items in this section
  • 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

    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

    No items in this section
  • 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

    No items in this section
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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 Mahathma Gandhi University, Kottayam.

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