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Metric spaces and Complex Analysis

By Dr. AJITHA V   |   MAHATMA GANDHI COLLEGE IRITTY
Learners enrolled: 136
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 : Upcoming
Course Type : Core
Language for course content : English
Duration : 16 weeks
Category :
  • Mathematics
Credit Points : 5
Level : Undergraduate
Start Date : 10 Jan 2025
End Date : 25 Apr 2025
Enrollment Ends : 28 Feb 2025
Exam Date : 25 May 2025 IST
NCrF Level   : 5.5
Exam Shift :

Shift - II

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits



Course layout

Weeks Weekly Lecture Topics (Module Titles)

1 Day 1 Module 1 : METRIC SPACES AND EXAMPLES
Day 2 Module 2 : METRIC SPACES - MORE EXAMPLES      
Day 3 Module 3 : SEQUENCES IN METRIC SPACES
Day 4 Module 4 : CONVERGENCE OF SEQUENCES IN METRIC SPACES     
Day 5

2 Day 1 Module 5 : OPEN SETS
Day 2 Module 6 : FUNDAMENTAL PROPERTIES OF OPEN SETS
Day 3 Module 7 : CLOSED SETS
Day 4 Module 8 : CANTOR SET  AND CLOSURE OF A SET
Day 5

3 Day 1 Module 9 :  BOUNDARY OF A SET AND DENSE SET
Day 2 Module 10 : THEOREMS ON OPEN AND CLOSED SETS
Day 3 Module 11: SUBSPACES IN METRIC SPACES       
Day 4 Module 12 : SEPARABLE SPACES
Day 5
4 Day 1 Module 13 : CONTINUITY
Day 2 Module 14: UNIFORM CONTINUITY
Day 3 Module 15: BAIRE'S THEOREM
Day 4 Module 16: THEOREMS ON CONTINUITY AND UNIFORM CONTINUITY  
Day 5
5 Day 1 Module 17: HOMEOMORPHISM
Day 2 Module 18 : CONNECTEDNESS
Day 3 Module 19 :  ROPERTIES OF COMPLEX NUMBERS
Day 4 Module 20 : FURTHER PROPERTIES OF COMPLEX NUMBERS       
Day 5

6 Day 1 Module 21 : POLAR AND EXPONETIAL FORM 
Day 2 Module 22 : FUNCTIONS OF A COMPLEX VARIABLE
Day 3 Module 23 : LIMIT OF  FUNCTIONS OF A COMPLEX VARIABLE
Day 4 Module 24 : POINT AT INFINITY
Day 5
7 Day 1 Module 25 : CONTINUITY OF FUNCTIONS OF A COMPLEX VARIABLE
Day 2 Module 26 : MAPPINGS
Day 3 Module 27 : DIFFERENTIATION OF  FUNCTIONS OF A COMPLEX VARIABLE
Day 4 Module 28 : CAUCHY-RIEMANN EQUATIONS - I
Day 5
8 Day 1 Module 29 : CAUCHY-RIEMANN EQUATIONS - II
Day 2 Module 30 : ANALYTIC FUNCTIONS
Day 3 Module 31 : FURTHER PROPERTIES OF ANALYTIC FUNCTIONS         
Day 4
Day 5

9 Day 1 Module 32 : EXPONENTIAL FUNCTIONS
Day 2 Module 33 : TRIGONOMETRIC FUNCTIONS
Day 3 Module 34 : LOGARITHMIC FUNCTIONS
Day 4
Day 5

10 Day 1 Module 35 : HARMONIC FUNCTIONS
Day 2 Module 36 : MAPPINGS BY ELEMENTARY FUNCTIONS  
Day 3 Module 37 : THEOREMS ON LIMITS AND DIFFERENTIATION FORMULAE 
Day 4  
Day 5  

11 Day 1 Module 38 : ELEMENTARY FUNCTIONS - I
Day 2 Module 39 : ELEMENTARY FUNCTIONS - II
Day 3 Module 40 : DEFINITE INTEGRALS
Day 4
Day 5

12 Day 1 Module 41 : CONTOURS
Day 2 Module 42 : CONTOUR INTEGRALS
Day 3 Module 43 : CAUCHY-GOURSAT THEOREM
Day 4
Day 5  

13 Day 1 Module 44 : CAUCHY'S INTEGRAL FORMULA
Day 2 Module 45 : LIOVILLE'S THEOREM
Day 3 Module 46 : SEQUENCES AND SERIES -I
Day 4
Day 5

 15 Day 1 Module 47 : SEQUENCES AND SERIES -II
Day 2 Module 48 : TAYLOR SERIES
Day 3 Module 49 : LAURENT SERIES
Day 4
Day 5

 16 Day 1 Module 50 : POWER SERIES
Day 2 Module 51: FURTHER PROBLEMS ON POWER SERIES                 
Day 3
Day 4
Day 5

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

Course Completion  will carry 70% weightage of end term exam and 30% weightage of internal assessments. A minimum 40% in each is required to qualify for the Course Completion Certificate.


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