Gautam Bharali

               Department of Mathematics

                 Indian Institute of Science

                 Bangalore 560012


Home Education Research Publications Abridged CV Miscellanea Teaching



  • Lecture hours

    Mondays, Wednesdays and Fridays 11:00 a.m.–12:00 noon

  • All about this course (PDF)

  • Recommended books

    Walter Rudin, Principles of Mathematical Analysis, 3rd Ediition, McGraw-Hill International Editions, 1976.

    Terence Tao, Analysis–I, 3rd Ediition, TRIM Series, Hindustan Book Agency, 2014.

    Terence Tao, Analysis–II, 3rd Ediition, TRIM Series, Hindustan Book Agency, 2014.

  • Teaching Assistant

    Anindya Biswas

    E-mail address: anindyababai1989@«...» (replace «...» by in the addresses)

    Office: R-22, Department of Mathematics

    Office hour: 5:30 to 6:30 p.m. on Thursdays

  • Documents

  • Syllabus (tentative: the list below will grow as the semester progresses the topics below comprise the syllabus of the final exam )

    Your lecture notes will cover all the material (except for those results assigned for self-study) in the syllabus. The occasional chapter references are to more extensive explanations, and refer to Rudin's Principles.

    Aspects of the theory of sets, relations and functions

    The natural numbers, the principle of mathematical induction, Peano arithmetic

    Number systems, the rational numbers, fields, ordered fields and the "usual order" on the rationals

    The least upper bound property, the real line, construction of the real line (Chapter 1: Appendix)

    The Archimedean property of the real line, complex numbers, Euclidean spaces, the Cauchy–Schwarz inequality and associated inequalities (the section The Complex Field in Chapter 1)

    Countable and uncountable sets, cardinality

    Metric spaces, open and closed sets in metric spaces and associated concepts

    Compact sets, the characterisation of compact subsets of Euclidean spaces

    Cantor sets, perfect sets (the section Perfect Sets in Chapter 2), connected sets

    Sequences and convergence

    Subsequences, subsequential limits, the limits of special sequences (the section Some Special Sequences in Chapter 3)

    Cauchy sequences, completeness, sufficient conditions for completeness

    Infinite series and their convergence, criteria for convergence

    Convergence tests for non-negative series, criterion for summability of the series of pth powers

    The extended real number system, limits at infinity, upper and lower limits

    Absolute convergence, the Ratio and Root Tests, conditional convergence, using convergence of series to derive limits of sequences

    Topics listed up to this point comprised the syllabus of the mid-term examination.

    The limit of a function: various equivalent definitions, the algebra of limits

    Continuous functions

    Continuity and compactness, attainment of extreme values, uniform continuity

    Continuity and connectedness, the intermediate-value theorem, and applications

    Left- and right-hand limits (as in the section Discontinuities in Chapter 1)

    Differentition in one variable

    Lagrange's mean value theorem and its applications

    The chain rule for differentiation in one variable

    Taylor's theorems: the approximation theorem and the mean value theorem

    Integration: motivation, the Riemann integral, and a characterisation of Riemann integrability

    Riemann integrability of continuous functions, functions with discontinuities

    The first and second Fundamental Theorems of Calculus

    Techinques: the change-of-variable formulas for the Riemann integral, integration by parts

    Sequences of functions, examples and motivations for uniform convergence

    Uniform convergence and Riemann integration

    The normed linear spaces Cb(X;R) and Cb(X;C) and their completeness

    Equicontinuity, compact sets in C(K;R): the Arzela–Ascoli Theorem

    The Weierstrass approximation theorem (Note: we omit the Stone–Weierstrass theorems.)

    Vector-valued functions in several real variables: limits and continuity

    Matrix representations of linear transformations, the matrix norm

    The meaning of differentiation of functions in several variables: the total derivative

    Partial derivatives, matrix representation of the total derivative

    The chain rule for the total derivative, applications of the chain rule

  • Announcements

    Nov. 30: The venue of the end-semester exam (on December 4) is Lecture Hall 4.

    Nov. 23: Anindya will hold office hours on Thursday, November 29, at the usual time and place.

    Nov. 12: The venue of the last make-up lecture (on November 17) is Lecture Hall 1.

    Nov. 16: The end-semester exam is scheduled for 9:00 a.m. on December 4. Venue: TBA.

    Nov. 12: The last make-up lecture will be on November 17. Time: 4:15 to 5:45 p.m. Venue: TBA.

    Oct. 20: There will be no classes on the week beginning October 22 since the instructor will be away on an academic visit. Classes will resume on October 29.

    Oct. 11: The third make-up lecture, which was announced on October 5, has been RESCHEDULED. It is still scheduled on October 13, but the new time for it is 4:00 to 5:30 p.m. Venue: Lecture Hall 1.

    Oct. 5: The third make-up lecture (to make up for regular lectures lost to holidays) will be on October 13. Time: 10:00 to 11:30 a.m. Venue: TBA.

    Oct. 5: There will be no lecture on October 8 owing to a holiday at the Institute.

    Sep. 26: The venue of the mid-term examination is Lecture Hall 1, Department of Mathematics.

    Sep. 20: There will be a lecture on September 21 at the usual place and time, even though the 21st is a holiday at the Institute.

    Sep. 19: There will be no classes during the mid-term examination week: i.e., September 24 to 28. However, Anindya will have office hours on September 27 (Thursday) as usual.

    Sep. 7: The first two of the make-up lectures (to make up for regular lectures lost to holidays) will be on September 8 and 15. Time and venue: 10:00 to 11:30 a.m., Lecture Hall 1.

    Sep. 1: The mid-term exam is scheduled for 10:00 a.m. on September 29. Venue: to be announced.

    Aug. 17: There will be a lecture on August 24 at the usual place and time, even though the 24th is a holiday at the Institute.

  • Homework assignments

    Homework 13

    Homework 12

    Homework 11

    Homework 10

    Homework 9

    Homework 8

    Homework 7

    Homework 6

    Homework 5

    Homework 4

    Homework 3

    Homework 2

    Homework 1

  • Quiz solutions

    The solution to Quiz 6

    The solution to Quiz 5

    The solution to Quiz 4

    The solution to Quiz 3

    The solution to Quiz 2

    The solution to Quiz 1


  • UNDERGRADUATE ANALYSIS & LINEAR ALGEBRA (UM101)   [Autumn 2013, Autumn 2015, Autumn 2017]


  • ANALYSIS–II: MEASURE AND INTEGRATION (MA222)  [Spring 2012, Spring 2017 ]

  • COMPLEX ANALYSIS (MA224)  [ Spring 2016 ]

  • TOPICS IN COMPLEX ANALYSIS (MA324)  [Spring 2014]

  • INTRODUCTION TO SEVERAL COMPLEX VARIABLES (MA328-329)  [experimentally as a "topics course" (MA329) in Autumn 2014 ]


Page last updated on November 30, 2018