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Metric Spaces and Complex Analysis

By Dr. AJITHA V   |   MAHATMA GANDHI COLLEGE IRITTY
Learners enrolled: 829
Learners having idea of fundamental Mathematics can easily understand the fundamentals of functions of a complex variable, metric spaces, and various theorems like Cantor’s theorem, Banach Fixed point Theorem, Cauchy-Riemann equations, Cauchy-Goursat theorem, Cauchy integral formula, Liouville’s theorem and the fundamental theorem of algebra.
The main objectives are 
To understand the concept of a metric space, to familiarize the ideas of open and closed sets, to learn the concept of continuity, homeomorphism and connectedness, to provide a foundation for more advanced courses in Mathematical analysis, to provide a new perspective on many of the ideas studied in Real Analysis, to study the techniques of complex variables and functions together with their derivatives, Contour integration and transformations and  developing a clear understanding of the fundamental concepts of Complex Analysis 

Summary
Course Status : Completed
Course Type : Core
Duration : 15 weeks
Category :
  • Mathematics
Credit Points : 4
Level : Undergraduate/Postgraduate
Start Date : 10 Jan 2023
End Date : 30 Apr 2023
Enrollment Ends : 15 Mar 2023
Exam Date :

Page Visits



Course layout

Weeks Weekly Lecture Topics (Module Titles)

1 Day 1 Module 1 : METRIC SPACES AND EXAMPLES
Day 2 Module 2 : SEQUENCES IN METRIC SPACES
Day 3 Module 3 : OPEN SETS 
Day 4
Day 5

2 Day 1 Module 4 : FUNDAMENTAL PROPERTIES OF OPEN SETS 
Day 2 Module 5 : CLOSED SETS
Day 3 Module 6 :CANTOR SET  AND CLOSURE OF A SET
Day 4
Day 5

3 Day 1 Module 7 : BOUNDARY OF A SET AND DENSE SET
Day 2 Module 8 : THEOREMS ON OPEN AND CLOSED SETS
Day 3 Module 9 : SEPARABLE SPACES
Day 4
Day 5
4 Day 1 Module 10 : CONTINUITY
Day 2 Module 11 : UNIFORM CONTINUITY
Day 3 Module 12 : BAIRE'S THEOREM
Day 4
Day 5

5 Day 1 Module 13 : HOMEOMORPHISM
Day 2 Module 14 : CONNECTEDNESS
Day 3 Module 15 : PROPERTIES OF COMPLEX NUMBERS
Day 4
Day 5

6 Day 1 Module 16 : POLAR AND EXPONETIAL FORM 
Day 2 Module 17 : FUNCTIONS OF A COMPLEX VARIABLE
Day 3 Module 18 : LIMIT OF  FUNCTIONS OF A COMPLEX VARIABLE
Day 4
Day 5

7 Day 1 Module 19 : POINT AT INFINITY
Day 2 Module 20 : CONTINUITY OF FUNCTIONS OF A COMPLEX VARIABLE
Day 3 Module 21 : MAPPINGS
Day 4
Day 5

8 Day 1 Module 22 : DIFFERENTIATION OF  FUNCTIONS OF A COMPLEX VARIABLE
Day 2 Module 23: CAUCHY-RIEMANN EQUATIONS - I
Day 3 Module 24 : CAUCHY-RIEMANN EQUATIONS - II
Day 4
Day 5

9 Day 1 Module 25 : ANALYTIC FUNCTIONS
Day 2 Module 26 : EXPONENTIAL FUNCTIONS
Day 3 Module 27: TRIGONOMETRIC FUNCTIONS
Day 4
Day 5

10 Day 1 Module 28 : LOGARITHMIC FUNCTIONS
Day 2 Module 29 : HARMONIC FUNCTIONS
Day 3 Module 30 : DEFINITE INTEGRALS
Day 4  
Day 5  

11 Day 1 Module 31 : CONTOURS
Day 2 Module 32 : CONTOUR INTEGRALS
Day 3 Module 33: CAUCHY-GOURSAT THEOREM
Day 4  
Day 5  

12 Day 1 Module 34 : CAUCHY'S INTEGRAL FORMULA
Day 2 Module 35 : LIOVILLE'S THEOREM
Day 3 Module 36 : SEQUENCES AND SERIES - I
Day 4  
Day 5  

13 Day 1 Module 37 :SEQUENCES AND SERIES - II
Day 2 Module 38 : TAYLOR SERIES
Day 3 Module 39 : LAURENT SERIES
Day 4 .
Day 5  

 14 Day 1 Module 40 : POWER SERIES
Day 2
Day 3
Day 4

Books and references

1. Satish Shirali and Harikishan L. Vasudeva, Metric Spaces, Springer Verlag, London, 2006.
2. S. Kumaresan, Topology of Metric Spaces, 2nd Ed., Narosa Publishing House, 2011.
3. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 2004.
4. James Ward Brown and Ruel V. Churchill, Complex Variables and Applications, 8th Ed., McGraw – Hill International Edition, 2009.
5. Joseph Bak and Donald J. Newman, Complex Analysis, 2nd Ed., Undergraduate Texts in Mathematics, Springer-Verlag New York, Inc., NewYork, 1997.

Instructor bio

Dr. AJITHA V

MAHATMA GANDHI COLLEGE IRITTY
Dr Ajitha V (Course Coordinator) has 27 years of experience both in UG and PG level.  She handled the topics Linear Algebra, Analytical Number Theory, Algebraic Number Theory, Theory of Numbers, Graph Theory, Commutative    Algebra and Abstract  Algebra under PG Level for the last 21 years.
She handled the topics Vector Analysis, General Topology , Complex Analysis, Real Analysis, Graph Theory, Informatics, Vedic Mathematics, Integral calculus, Differential Calculus, Differential Equations, Business Mathematics, Numerical analysis  under UG level last 27 years.
Awarded M Phil Degree from University of Calicut in 1990
She is the Content Editor in the development MOOC modules for the course Differential Calculus.
QP Setter for the post of lecturer in Mathematics for Kerala PSC. 
Member of the scrutiny of the Answer key for the post of lecturer in Mathematics for Kerala PSC.
Subject expert for the selection of the post of HSA Mathematics for Kerala PSC.
Mathematics - Board of Studies member
        UG -2003-05,   PG – 2005-07 & 2010-12, 2019-22.
Academic Council Member of Kannur University.
Convener, UGC Sponsored National seminar.
More than 15 publications in International Journals.
Awarded Ph.D. from the Kannur University in 2008 for thesis titled  Studies in Graph Theory-Labeling of Graphs under the guidance of Dr. Sr. Germina K A.
Developed study materials for Kannur University and University of Calicut for their School of Distance Education. 

Course certificate

Assessment/Assignment marks will be considered for Internal Marks and will carry 30 percent for overall Result.

End Term Exam- will have 100 questions and will carry 70 percent of overall Result.

*All students, who obtain 40% marks in in-course assessment and 40% marks in end-term proctored exam separately, will be eligible for certificate and credit transfer.



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