MATH-4200, MATHEMATICAL ANALYSIS I

# MATHEMATICAL ANALYSIS I

Time: Monday and Thursday, 10:00 to 11:50 PM.
Room: Lally 104
Instructor: Gregor Kovacic
Office: 420 Amos Eaton
Phone: 276-6908
E-mail: kovacg at rpi dot edu

• This document contains the list of topics to be presented in this course, a list of textbooks, and some comments and recommendations about these textbooks.
• Click on each topic title to download the notes for that topic.

## Topics

Click on each topic title to download the notes for that topic.

Number Systems: Integers, countable infinity, induction, rationals, irrationals, the ordered ring of real numbers, suprema and infima, the least upper bound and Archimedean properties, decimal representation, uncountable infinity of real numbers, basic inequalities, complex numbers, roots of unity.

Topology of the Real Axis and the Complex Plane: Basic set theory, Cartesian producs, Axiom of Choice, open and closed intervals, open and closed balls, open and closed sets, cluster points, interiors, closures, boundaries, bounded sets, connected sets, open subsets of the real line, Cantor set.

Limits: Functions, domains, ranges, onto and one-to-one functions, inverses, infinite sequences and their limits, convergence and divergence, Cauchy sequences, bounded sequences and cluster points, monotonic sequences, operations on limits, some special limits, limsup and liminf, completeness of real numbers, construction of the real numbers, limits of functions, left-hand and right-hand limits.

Continuity: Continuity at a point, limits and continuity, continuous functions, examples of continuous functions, operations on continuous functions, continuity of composite functions, continuous preimages of open and closed sets, compactness, Bolzano-Weierstrass and Heine-Borel theorems, uniform continuity, continuous images of compact intervals, Lipschits and Holder continuity, discontinuities, monotonic functions and their inverses.

Differentiation: Derivative at a point, slope of the tangent, differentiable functions, derivatives of elementary functions, differentiability and continuity, calculus of derivatives, derivatives of composite functions and the chain rule, mean value theorem, derivatives and monotonicity, derivatives of inverse functions, higher derivatives, maxima and minima of functions, lack of differentiability at a point, intermediate value theorem, L'Hospital's rule.

Rieman Integral: Riemann and Darboux sums, partitions and refinements, the Riemann integral, integrals of continuous functions, operations on integrals, integrals of the absolute value, integrals over adjacent intervals, mean value theorem, antiderivatives and the fundamental theorem of calculus, change of variable, integration by parts, integrability of piecewise-continuous functions, a nonintegrable function, logarithm and exponential functions, hyperbolic functions and their inverses, integration methods for rational functions, improper integrals, gamma function.

Numerical and Functional Series: Taylor's formula and Taylor series, Lagrange's and Cauchy's remainder, Taylor expansion of elementary functions, indefinite expressions and L'Hospital rule, numerical series, Cauchy's criterion, absolute and conditional convergence, addition and multiplication of series, functional sequences and series, pointwise and uniform convergence, Weierstrass test, integration and differentiation of functional series, power series and radius of convergence, complex exponentials, Weierstrass Approximation Theorem.

Class notes from the Fall semester of 2015 are deposited here.
Class notes from the Fall semester of 2014 are deposited here.
Class notes from the Fall semester of 2013 are deposited here.
Here is another set of notes, written up by Joshua Sauppe.

## Textbooks

The following textbooks contain (some of the) material presented in this course:

T. M. Apostol, Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, Wiley.
T. M. Apostol, Mathematical Analysis, Second Edition, Addison-Wesley.
A. Browder, Mathematical Analysis: An Introduction, Springer-Verlag.
R. C. Buck, Advanced Calculus, Waveland.
R. Courant, Differential and Integral Calculus, Vol. 1, Springer-Verlag.
R. Courant and F. John, Introduction to Calculus and Analysis, Vol. 1, Springer-Verlag.