27210 - Mathematical Analysis 1 (M-Z)

Learning outcomes

At the end of the course, the student will have basic notions of infinitesimal and integral calculus, simultaneously developing the habit of scientific reasoning and a sensitivity to the analysis of mathematical models, especially through the study of the asymptotic development of functions. Furthermore, he will be able to carry out a detailed study of functions in a variable, of sequences and series, both numerical and of functions.

Course contents

MATHEMATICAL LOGIC
Hints of logic. Symbols and types of proofs.

NUMERICAL SETS
Definition of the numerical sets N,Z,Q,R. Total ordering of R. Existence of lower and upper bounds of a subset of R. Principle of induction. (Cardinality of a set, countability of rationals, uncountability of real numbers.)
Newton's binomial.

COMPLEX NUMBERS
Definition of complex number and its representation on the Gaussian plane. Real part, imaginary part, modulus, conjugate and their properties. Operations on complex numbers. Trigonometric form and algebraic form of a complex number, transition from one form to another. (Euler's formula and exponential form of a complex number.) Powers and roots of complex numbers: De Moivre's formula and n-simal roots of unity. Algebraic equations and fundamental theorem of algebra.

FUNCTIONS
Real functions of real variable. Symmetries, periodicity, monotonicity, invertibility. Elementary functions, definition (and existence) of the n-th root of a non-negative number, definition of power with real exponent.

LIMITS
ELEMENTS OF TOPOLOGY OF THE LINE: Absolute value and its properties. Intervals, open, closed, bounded, compact sets. Accumulation points and isolated points.
LIMITS: definition of limit, uniqueness of the limit, sign permanence theorem, comparison theorems, algebra of limits.
SEQUENCES: definition of convergent, divergent, indeterminate succession. Theorem on the limits of monotone sequences. Definition of Neper's number.

CONTINUOUS FUNCTIONS: definition of continuity, different types of discontinuity. Main theorems: Zeros Theorem, Intermediate Value Theorem and Weierstrass Theorem. Uniformly continuous functions. Heine-Cantor Theorem. Continuity of elementary functions. Continuity of the inverse function of a continuous and invertible function defined on an interval.

DERIVATIVES
Definition of derivative. Algebra of derivatives. Derivative of the composite function. Derivative of the inverse function. Differentiability of elementary functions. The Theorems of Rolle, Lagrange, Cauchy and De L'Hopital. Taylor formula with Peano, integral and Lagrange remainder.

INTEGRALS
Primitives. Definition of Riemann integral of a bounded function. Riemann integrability of continuous (and bounded) functions. Properties of the integral. Integration methods: integration by parts, by substitution, method of simple fractions (with real or complex roots of arbitrary multiplicity). Mean Theorem and Fundamental Theorem of Integral Calculus. Improper integrals.

SERIES
Definition of convergent, divergent or indeterminate series. Necessary condition for the convergence of a series. Series with non-negative terms (ratio, root and condensation criteria). Geometric series, harmonic series and generalized harmonic series. Absolute convergence. Leibniz criterion (for alternating sign series).

ORDINARY DIFFERENTIAL EQUATIONS
First order ordinary differential equations with separable variables. First order linear differential equations. Second order linear equations with constant coefficients (homogeneous and complete). Variation of arbitrary constant method and similarity method. Solving the Cauchy problem for the previous classes of equations. Linear differential equations of order n with variable coefficients. Contractions theorem. Existence and uniqueness theorem of solutions of first order differential equations.

SEQUENCES AND SERIES OF FUNCTIONS
Uniform convergence of a succession of functions. (Completeness of the set of continuous functions defined on a compact with respect to the distance associated with the uniform norm.) Passage to the limit below the integral sign. Total convergence of a series of continuous functions.

Readings/Bibliography

• C. Canuto, A. Tabacco. Analisi matematica 1, Pearson -- (theory and exercises)
• G.C. Barozzi, G. Dore, E. Obrecht. Elementi di analisi matematica 1, Zanichelli -- (thoery)
• P.Marcellini, C.Sbordone. Analisi Matematica uno, Liguori Editore --(theory)
• C.Pagani, S.Salsa. Analisi matematica 1, Zanichelli -- (theory and exercises)
• E.Giusti. Analisi matematica 1, Bollati Boringhieri -- (theory and exercises)
• S.Salsa, A.Squellati. Esercizi di Analisi matematica 1, Zanichelli -- (exercises)

Teaching methods

Lectures and exercises in class.

Assessment methods

Written exam on the exercises part

The test consists in solving exercises on the topics of the course: study of functions, complex numbers, calculation of limits, study of the convergence of numerical series and functions, solving integrals and ordinary differential equations.

Theory exam

The theory exam consists of a compulsory written part and an optional oral part. You can access the theory exam only after passing the exam on the exercises part with a grade greater than or equal to 18.

The detailed exam methods will be explained in class and published on Virtuale.

Teaching tools

Teaching resources on Virtuale

Office hours

See the website of Francesca Colasuonno