Analysis I


Homework I (due 17 August 2004)
Homework II (due 24 August 2004)
Homework III (due 31 August 2004)
Homework IV (due 7 September 2004)
Homework V (due 14 September 2004)
Homework VI (due 21 September 2004)
Homework VII (due 28 September 2004)
Homework VIII (due 19 October 2004)
Homework IX (due 2 November 2004)
Homework X (due 9 November 2004)
Homework XI (due 23 November 2004)
Homework XII (due 30 November 2004)

Final Exam

Recommended reading

Apostol, T.M., Mathematical analysis, Addison-Wesley.
Artin, E., The Gamma function, Holt, Rinehart and Winston.
Bromwich, T.J.I'a, An introduction to the theory of infinite series, MacMillan & Co. Ltd.
Dedekind, R., Essays on the theory of numbers, Dover Publications Inc.
Euclid, The Elements (Translated with Introduction and commentary by Sir Thomas L. Heath), Dover Publications Inc. Some further comments.
Hardy, G.H., Pure Mathematics, Cambridge University Press.
Rudin, W., Principles of mathematical analysis, McGraw-Hill.
Simmons, G.F., Introduction to topology and modern analysis, McGraw-Hill.
Whittaker, E.T. and Watson G.N., A course of modern analysis, Cambridge University Press.

Topics discussed (loosely)

Construction of real numbers by Dedekind cuts, completeness; rational, irrational and transcendental numbers, Liouville's theorem on approximation of algebraic numbers.

Sequences and series, convergence, conditional and absolute; open, closed and compact sets, Heine-Borel theorem; Abel's inequality, Dirichlet's test, Cauchy's root test, D'Alembert's ratio test, other comparisons, deMorgan's theorem applied to hypergeometric series; double series, Pringsheim's theorem.

Continuous functions, intermediate value theorem, uniform convergence (for functions), preservation of continuity under uniform convergence, Weierstrass M-test, application to power series.

Abel's theorem on continuity up to the circle of convergence, Abel's theorem on products of series, sums and products of power series, composition, inversion, differentiation and integration of functions given by power series, the principle of analytic continuation, exponential, logarithmic and trigonometric functions, periodicity of the exponential function.

Infinite products, Euler product for the Riemann zeta function, Euler's theorem on the divergence of the series of reciprocals of prime numbers.

Gamma functions, characterisation in terms of functional equation and convexity of its logarithm (see Rudin's book).

Functions on a closed interval-maximum and minimum of a continuous function, Riemann integral of a bounded function, first and second (Bonnet's) mean value theorems for integrals, improper integrals of the first and second type, absolute convergence, comparison tests, Chartier's test.

Integration of monotone functions, functions of bounded variation, their characterization as difference of increasing functions, thier integrability.

Lebesque's theorem characterizing Riemann integrable functions.

Double integrals and repeated integrals.

Example of a continuous function which is nowhere differentiable.

Orthonormal systems, trigonometric series, Fourier coefficients, Bessel's inequality, the Dirichlet kernel, partial sums of Fourier series in terms of the Dirichlet kernel, localization theorem, Fejer's theorem, Parseval's theorem, approximation of Riemann integrable functions by continuous functions (in the L^2 norm).

Weierstrass approximation theorem, algebras of functions, Stone-Weierstrass theorems for real and complex-valued functions.


Biographies of Mathematicians (University of St. Andrews, Scotland).
Works of Hamilton and Riemann.
Euclid's Elements online!
Which famous mathematicians were born today?