Teaching - Università Roma Tre

AM110 - MATHEMATICAL ANALYSIS 1 (objectives)

Code

20410405

Language

ITA

Type of certificate

Profit certificate

Credits

9

Scientific Disciplinary Sector Code

MAT/05

Contact Hours

48

Exercise Hours

42

Type of Activity

Basic compulsory activities

Teacher	CHIERCHIA LUIGI (syllabus) PART 1: The set of real numbers and its main subsets • Sets, relations and functions. • Axioms of real numbers. • Elementary properties of ordered fields. • Symmetric sets and functions. Absolute value and distance. • Natural numbers. Subtraction in N; principle of well-ordering and its consequences. • Sequences and recursion theorem (optional proof). Recursive definition of sums, products and powers. • N^th powers, geometric sum and formula for a^n-b^n. Newton's binomial formula. • Finite and infinite sets. • Rational numbers. The rationals are countable. Gauss lemma. • Least upper bound and greatest lower bound. Elementary consequences of the completeness axiom on integers. • Roots. Powers with rational exponent. • Monotone functions. PART 2: Theory of limits • The extended real system R*. Intervals and neighbourhoods. • Internal, isolated, accumulation points. General definition of limit. Uniqueness of the limit. • Sign permanence theorem. Comparison theorems. • Side limits and monotone functions. • Algebra of finite limits. Extended limit algebra. • Some notable limits of sequences. • The number of Nepero. • Bridge theorem and characterisation of the sup / inf by sequences. • Continuity: general considerations; theorem of existence of zeros. Intermediate value theorem. • Classification of discontinuities. • Limits for compound functions. • Limits for inverse functions. • A continuous and strictly monotone function on an interval admits a continuous inverse. • Logarithms. • Notable limits (exponential and logarithms). PART 3: Series • Numerical series: Elementary properties of series. Comparison criteria. • Decimal expansions. • Convergence criteria for series with positive terms. • Criteria for series with real terms (Abel-Dirichlet, Leibniz). • Exponential series. Irrationality of e. Speed of divergence of the harmonic series. • Properties of trigonometric functions (in particular proof of the cosine addition theorem). • Periodic functions. Monotonic properties of trigonometric functions. • Inverse trigonometric functions. (reference books) Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su R McGraw-Hill Education Collana: Collana di istruzione scientifica Data di Pubblicazione: giugno 2019 EAN: 9788838695438 ISBN: 8838695431 Pagine: XI-374 Formato: brossura https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte Exercise texts: Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000 Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Dates of beginning and end of teaching activities	From to
Delivery mode	Traditional At a distance
Attendance	not mandatory

Teacher	MATALONI SILVIA (syllabus) PART 1: The set of real numbers and its main subsets • Sets, relations and functions. • Axioms of real numbers. • Elementary properties of ordered fields. • Symmetric sets and functions. Absolute value and distance. • Natural numbers. Subtraction in N; principle of well-ordering and its consequences. • Sequences and recursion theorem (optional proof). Recursive definition of sums, products and powers. • N^th powers, geometric sum and formula for a^n-b^n. Newton's binomial formula. • Finite and infinite sets. • Rational numbers. The rationals are countable. Gauss lemma. • Least upper bound and greatest lower bound. Elementary consequences of the completeness axiom on integers. • Roots. Powers with rational exponent. • Monotone functions. PART 2: Theory of limits • The extended real system R*. Intervals and neighbourhoods. • Internal, isolated, accumulation points. General definition of limit. Uniqueness of the limit. • Sign permanence theorem. Comparison theorems. • Side limits and monotone functions. • Algebra of finite limits. Extended limit algebra. • Some notable limits of sequences. • The number of Nepero. • Bridge theorem and characterisation of the sup / inf by sequences. • Continuity: general considerations; theorem of existence of zeros. Intermediate value theorem. • Classification of discontinuities. • Limits for compound functions. • Limits for inverse functions. • A continuous and strictly monotone function on an interval admits a continuous inverse. • Logarithms. • Notable limits (exponential and logarithms). PART 3: Series • Numerical series: Elementary properties of series. Comparison criteria. • Decimal expansions. • Convergence criteria for series with positive terms. • Criteria for series with real terms (Abel-Dirichlet, Leibniz). • Exponential series. Irrationality of e. Speed of divergence of the harmonic series. • Properties of trigonometric functions (in particular proof of the cosine addition theorem). • Periodic functions. Monotonic properties of trigonometric functions. • Inverse trigonometric functions. (reference books) Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su R McGraw-Hill Education Collana: Collana di istruzione scientifica Data di Pubblicazione: giugno 2019 EAN: 9788838695438 ISBN: 8838695431 Pagine: XI-374 Formato: brossura https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte Testi di esercizi: Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000 Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010 Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su R McGraw-Hill Education Collana: Collana di istruzione scientifica Data di Pubblicazione: giugno 2019 EAN: 9788838695438 ISBN: 8838695431 Pagine: XI-374 Formato: brossura https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Dates of beginning and end of teaching activities	From to
Delivery mode	Traditional At a distance
Attendance	not mandatory