## "Babes-Bolyai" University of Cluj-Napoca Faculty of Mathematics and Computer Science

 Mathematical analysis (4)
 Code Semes-ter Hours: C+S+L Credits Type Section MO031 4 2+2+0 6 optional Matematică MO031 4 2+2+0 6 optional Matematică MO031 4 2+2+0 6 optional Matematică-Informatică MO031 4 2+2+0 6 optional Matematică-Informatică
 Teaching Staff in Charge
 Lect. FINTA Zoltan, Ph.D., fzoltan@math.ubbcluj.roAssoc.Prof. DIACONU Adrian, Ph.D., adiaconu@math.ubbcluj.ro
 Aims Getting to know the analitical representation of the functions. The theory of Fourier series.
 Content 1. The definition and the representation of classical elementary functions : the constructive definition of exponential functions of base e. The function ln and the logarithmic function of base a and exponential function of base a, respectively. The power function. The functions sin, cos, tg, ctg, arcsin, arccos, arctg and arcctg. Properties. Expansion of the functions sin and cos in infinite product. 2. Fourier's series : trigonometric Fourier series. Pointwise convergence and uniform convergence of trigonometric Fourier series. The theorems of Dini, Dirichlet - Jordan and Fejer. The Parseval - Liapunov's identity. Sequences of orthogonal functions. Orthogonal polynomials. Fourier series regarding orthogonal sequences. Convergence theorems. 3. Parametric integrals : the pointwise limit and the uniform limit of functions of several variables. Properties of the limit function. The existence test of the uniform limit. The uniform convergence of parametric improper integrals. Uniform convergence tests. Operation on parametric integrals ( passing to the limit, derivation, integration ). The Euler integrals of the first kind and of the second kind ( beta - function and gamma - function ). Properties.
 References l. Balazs M., Kolumban I.: Matematikai analizis, Dacia Konyvkiado, Kolozsvar-Napoca, 1978 2. Fihtenholt G.M.: Curs de calcul diferential si integral, II - III, Editura tehnica,Bucuresti, 1965 3. Luzin N.N.: Calcul integral, Editura tehnica, Bucuresti, 1955 4. Marusciac I.: Analiza matematica,II, Universitatea Babes-Bolyai, Cluj-Napoca, 1983
 Assessment Exam.