27991 - Mathematical Analysis T-1

Course contents

PROPERTIES OF REAL NUMBERS. 
LIMITS AND CONTINUOUS FUNCTIONS. Definition of a convergent and of a divergent sequence of real numbers. Theorems about limits of sequences: uniqueness of the limit, comparison theorems. The algebra of limits. Monotone sequences and their limits. The number e. Decimal representation of real numbers. Generalities about functions: composition of functions, invertible functions and inverse functions. Pecularities of real-valued functions of one real variable. Definition of a continuous function of one real variable. The Weierstrass theorem and the intermediate value theorem. Definition of limit of a real function of one real variable; generalization of results established for sequences. Continuity of the composition of two continuous functions and the theorem on the change of variable in a limit. One-sided limits. Monotone functions and their limits. Asymptotes. The inverse circular functions. The hyperbolic functions and their inverse functions. 
DIFFERENTIAL CALCULUS. Definition of a differentiable function and of derivative of a function. The algebra of derivatives. The chain rule. The mean value theorem and its application to study the monotonicity of a function. Higher order derivatives. Taylor's formula with Peano and Lagrange form of the remainder. Relative maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions. 
INTEGRAL CALCULUS. Definition of the Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem. Sufficient conditions of integrability. The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts. Piecewise continuous functions and propeties of their integrals. Improper integrals: definitions, absolute convergence, comparison theorem.
COMPLEX NUMBERS. Definition and operations on complex numbers. Algebraic form of a complex number, modulus and argument of a complex number, exponential form of a complex number. De Moivre formula, roots of a complex number, algebraic equations in C, the complex exponential function. 
SERIES. Series with real and complex terms. Definition of a convergent series. Absolute convergence of a series. Convergence criteria for numerical series. 
LINEAR DIFFERENTIAL EQUATIONS. Linear differential equations of first order: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Linear differential equations of second order with constant coefficients: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Generalization to variable coefficients and arbitrary order equations.

Readings/Bibliography

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, Zanichelli (2009)
An exercise book on functions of one  real variable, for example: M. Amar, A.M. Bersani: Esercizi di Analisi Matematica, Progetto Leonardo (2004)

Teaching methods

Lessons and exercises in classroom.

Assessment methods

Written and oral examination. The written examination may be replaced by two partial written examinations.

Links to further information

http://www.dm.unibo.it/~obrecht/

Office hours

See the website of Enrico Obrecht