00013 - Mathematical Analysis

Learning outcomes

The aim of the course is to provide the capability of the student to face both theoretical and practical problems in Mathematical Analysis, referring to the analysis of the behavior of a real function of a real variable, computation of definite integrals, development of a function in power series.

Course contents

The set R of real numbers.  The main subsets of R: Natural, Integers and Rational numbers. Completeness axiom. Archimedean property.  Sequences and Series. Limit of a sequence. Monotonic sequences and the number e. Cesaro Stolz Theorems. Geometric series. Series with positive terms and convergence tests. Series with alternating terms. Continuous fractions and infinite products.  Complex Nembers. Algebraic representation of a complex number. Tha complex plane. Trigonometric form. De Moivre formulas Real functions. Limits and elementary functions. Asymptotics and Landau symbols. Continuous functions. Bolzano theorem on intermediate value and Weierstrass theorem on maxima and minima. Derivatives. Theorems of Rolle, Lagrange, Cauchy and De l'Hopital. Graph of a function, extrema. Convex and concave functions. Inflexion points. Asymptotes. Taylor polynomials and series. Riemann integral. Fundamental thoerems of Calculus. Integration methods. Improper integrals Sequences and Series of functions. Simple and uniform convergence. Theorems on the exchange between limit of a sequence of functions and the integral. Powers series. Taylor series.   Ordinary Differential Equation. Introduction to elementary differential equation of first order: separable and linear.

Readings/Bibliography

Daniele Ritelli. Lezioni di Analisi Matematica. Esculapio  
Marco Bramanti. Esercitazioni di Analisi Matematica 1. Esculapio  

David Brannan. A First Course in Mathematical Analysis. Cambridge University Press  
R.P. Burn. Numbers and functions. Steps into Analysis. Third edition. Cambridge University Press

Teaching methods

Lessons ex cathedra using also video beamer. Homework. Computer algebra will also be employed to support thoretical arguments.

Assessment methods

Written examination of 2 hours, where is possible to use calculators and books. The exam is completed by an oral examination if the written examination is satisfactory. The aim of the exam is to detect the capability of the student to face both theoretical and practical problems in Mathematical Analysis. The written examination can be divided, for the first call, in to two partial examinantion and is composed by multiple choice questions and solution of exercises.  

Teaching tools

Video beamer and blackboard.  Computer algebra to illustrate important topics.

Links to further information

http://www.ams.org/mathscinet/MRAuthorID/618511

Office hours

See the website of Daniele Ritelli