# MATHEMATICAL ANALYSIS - FIRST COURSE

# MATHEMATICAL ANALYSIS - FIRST COURSE

Elementary properties of sets and their operations. Basic knowledge of logic,

and of the propositional calculus. Relations,

equivalence relations, quotient space.

Functions, domain, codomain, image, preimage. Injective, surjective and bijective functions. Invertibility of a function.

Elementary inequalities, absolute value, positive and negative parts. Roots of polynomials of degre one and two.

To provide a solid theoretical basis and ability in mathematical reasonings, on the base of logical-deductive arguments. To develop skills and ability to solve the typical problems of a first course in Mathematical Analysis.

The natural numbers. Induction.

The sum of the first n squares. The sum of the first n cubProperties of the limiti: restriction, localization,

union, composition. Continuity of functions

at one point and on a set. Examples of discontinuous

functions.

es, etc.

The sum of the first n odd numbers. Anagrams, anagrams with repetitions. Binomial coefficients. Multinomial coefficients.

Bernoulli inequalities I and II.

The rational numbers. Countability of rational numbers. The real numbers.

The square root of 2 is not rqtional. Uncountability of real numbers.

Density of rational numbers, density of diadic numbers. Axiom of completeness.

Supremum, infimum, maximum, minimum of a nonempty subset of the real numbers. Series: series with positive terms. Geometric series. Mengoli series. Convergence of

$\sum 1/k^2$.

Comparison theorem. Comparison theorem definitely. Divergence of the monic series. The number e.

Remarks on the mononotonicity of the partial sums, related to the notion of limit. Area of the Cantor set in the unit square.

Convergence criteria: ratio and n-square root.

An example where one can apply the square root criterion but not the ration criterion.

If the series

\sum a_n with positive terms

converges,

then for any

\epsilon >0

the sequence

(a_n) is definitely

smaller than \epsilon.

Cauchy condensation

theorem. e is irrational.

Limsup and liminf of a sequence of real numbers.

sequences.

Finite or infinite limit of a sequence of real numbers. Uniqueness of the limit. A monotote sequence always admits the limit. Theorem of the permanence of the sign for sequences.

The sequence $(1 + 1/n)^n$: incresing monotonicity and existence of the limit equal to e. A first estimate of the factorial. Ratio and square root criteria, ratio-square root criterion for sequences. Subsequences. A real sequence always admits a converging subsequence to the liminf and to the limsup. Important limits. Summation by parts formula of Abel. Bolzano-Weierstrass Theorem.

Leibnitz criterion. Rearrangements of a series.

Dirichlet theorems on rearrangements of an

absolutely converging series.

Leibnitz criterion. Rearrangements of a series.

Dirichlet theorems on rearrangements of an

absolutely converging series.

Functions of one real variable:

Limits of functions

defined on subsets of the reals.

Equivalence between te metric definition and the

sequential definition.

Finite and infinite limits.

Examples of existence and nonexistence of the limits.

Limit of the sum, of the product etc.

Limit of the product of an infinitesimal function

and a bounded function.

A compact subset of R is closed and bounded,

and vice-versa.

Properties of the limiti: restriction, localization,

union, composition. Continuity of functions

at one point and on a set. Examples of discontinuous

functions.

Further properties of continuous functions. Continuity

of composition.

A function which is discontinuous only on Q. Weierstrass and intermediate value theorem.

monotone functions. continuity, limits. Uniform continuity.

Differentiability. Local and global maxima and minima. Global study of the graph of a function.

Convex functions of one real variable. The Riemann integral. Methods of integration. Generalized integrals.

Linear ordinary differential equations with constant coefficients. Some notions on continuity and differentiability

of functions of several variables.

1. G. Gilardo,

Analisi Matematica 1,

Mc-Graw Hill

2. E. Acerbi, G. Buttazzo,

Primo Corso di Analisi Matematica,

Pitagora Editrice, Bologna

3. J.P. Cecconi, G. Stampacchia,

Analisi matematica volume 1. Funzioni di una variabile,

Liguori Editore, Napoli

4. J.P. Cecconi, L. Piccinini, G. Stampacchia,

Esercizi e problemi di analisi matematica uno.

Liguori Editore, Napoli

TESTI DI CONSULTAZIONE

1. T.M. Apostol,

Calcolo,

Volume primo, Analisi 1,

Boringhieri Editore

2. F. Conti, P. Acquistapace, A. Savojni,

Analisi Matematica. Teoria e applicazioni,

Mc Graw-Hill

3 G. Prodi,

Analisi Matematica,

Boringhieri Editore

Traditional lectures and exercises. The pdf files of the lectures are available online to the

students after each lecture, see the webpage, at the address https://www3.diism.unisi.it/~bellettini/analisi1.html

The exam consists of a written part and of an oral part. In the written part (which lasts typically 2.30 or 3.00 hours) the students must solve exercises on the series, sequences, limits of functions, on the graph of a function, on integrals and on the final part of the course.

The oral part,

typically in the days immediately after the written part, is concerned with theoretical knowledge of the arguments on which the written part is based. Students are required to sign up online both at the written part of the exam, and at the oral part of it. At the written part of the exam, at each student will be given some sheets containing the text of the exercises, which the students are supposed to fill and give back.

Two office hours:

due to the actual epidemic situation, the office hours is scheduled online. Please, write at

bellettini@diism.unisi.it

webpage: http://www3.diism.unisi.it/~bellettini/

lessons online:

http://meet.google.com/wwt-ssft-wdk