Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMA1018	Topics of Mathematical Analysis III (for teachers education)	

Section	Semester	Hours: C+S+L	Category	Type	
Didactic Mathematics	4	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Prof. COBZAS Stefan, Ph.D., scobzas@math.ubbcluj.ro	
Aims	A deeper study of some properties of the Riemann and Riemann-Stieltjes integrals with applications to integrals depending on parameters. An introduction to Fourier series with applications to the calculation of the sums of some numerical series. The proof of some deep results on the a.e. differentiability of monotone and of absolutely continuous functions.	
Content	1.	The second mean value theorem for integrals and applications
The 2nd mean value theorem for Riemann and Riemann-Stieltjes integrals. The 2nd mean value theorem for the Lebesgue integral.  Abel&#8217;s and Dirichlet&#8217;s criteria on the convergence of improper integrals. The generalized formula of integration by parts and the transcendence of the numbers e and pi. The arithmetic-geometric  mean and Gauss formula.

 2. Arzela&#8217;s dominated convergence theorem and applications
Arzela&#8217;s theorem for Riemann integral versus the Lebesgue dominated convergence theorem. Applications  to passing to limit under the integration sign.

3.  Integrals with parameters &#8211; proper and improper. 
Applications of the Arzela&#8217;s theorem to integrals depending on parameters &#8211; continuity, derivability, integrability.  Examples of calculation of some improper integrals.  Improper integrals depending on parameters, the calculation of some remarkable improper integrals.  The applications of the Theorem of residues  to the calculation of improper integrals. Euler&#8217;s&#8217; Beta and Gamma functions. 

4. Fourier series and applications to the calculation of sums of some series 
The trigonometric system, the complex form of the trigonometric system, Fourier coefficients, Riemann&#8217;s Lemma. Fourier series expansion:  the completeness of the trigonometric system, the convergence in L2 ,  the Fourier series development of some elementary functions.
The uniform convergence of Fourier series with applications to the calculation of the sums of series. The pointwise convergence of Fourier series.  A short historical overview of the evolution of Fourier series. Criteria of pointwise convergence: Dini and Lipschitz, Dirichlet-Jordan, examples and applications. The Fourier transform and applications to the calculation of some integrals. The application of the Theorem of Residues to the calculation of some integrals and of some Fourier transforms.
 
5.  Absolutely continuous functions &#8211; a.e. derivability and their relation with the Lebesgue integral 
Vitali&#8217;s  covering theorem. The a.e. derivability of monotone functions, the a.e. derivability of functions with bounded variation.. The differentiability of function series &#8211; Dini&#8217;s theorem.  Lebesgue&#8217;s density theorem. The a.e. derivabilitity of absolutely continuous functions. The relation with the  primitive (antiderivative).  

Bibliography	1. Cobzas, S., Analiza matematica &#8211; Calculul diferential, Presa Universitara Clujeana, 
Cluj-Napoca 1997
2. Fihtenholt, G. M., Curs de calcul diferential si integral, Volumele 2 si 3, 
Editura Tehnica, Bucuresti 1963   
3. Hewitt, E.; Stromberg, K. Real and abstract analysis, Springer, Berlin 1965
4. Nicolescu, M., Analiza matematica, Vol. 2, Editura Tehnica, Bucuresti 1958
5. Rudin, W. Principles of mathematical analysis , McGraw, Ny 1976
6. Stein, E. M.; Shakarchi, R., Real analysis, Princeton University Press, 
Vol. 2, Fourier analysis &#8211; An introduction ;  2003
 Vol. 3, Measure theory, integration and Hilbert spaces,  2005 
7. Zuilly, Cl., Elements d&#8217;analyse pour aggregation, Masson, Paris 1995
8. Duca, D; Duca, E., Analiza matematica &#8211; Culegere de probleme, Editura GIL, Zalau 1999
9. Trif, T., Probleme de calcul diferential si integral in Rn, Casa Cartii de Stiinta, Cluj-Napoca 2003
10. Problemes de preparation a l&#8217;agregation mathematiques, Vol. 3. Analyse; Vol. 4.   Integration et series de Fourier, Ellipses, Paris 1999
	
Assessment	Written examination 50%; written control  tests 25%: talks presented at seminars 25% students.