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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 MeanValue 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 MeanValue
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: 1.11.3, Chapter 7 : Section 3 : 3.1 – 3.3)
COURSE OUTLINE
No  Topics  Learning Outcomes  Hours 
1  Harmonic Functions – Definitions and Basic Properties
The MeanValue 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 MeanValue 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 2125 

The Differential Equation 
3 
September 2830 

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:
 https://www.youtube.com/watch?v=v9nyNBLCPks
 https://www.youtube.com/watch?v=b5VUnapuqs
 https://www.youtube.com/watch?v=b5VUnapuqs
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 MeanValue 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

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

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

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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