253174132-Culegere-Exercitii-de-Analiza-Reala-Si-Complexa.pdf

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,. ' 'P --_ .. - [ ) . ': . Prof. dr. Gheorghe Opri~an r Lect dr. Antonela Toma As: -drd:"Cristhla Anton ~ u - , ~ " ~~ ;.. .'. , . ... .• _ !~ ... ... \.. ." . h ): .. jJ'~ \. .1lJ ... ".L.JL ...... '_.~ ~ ;.l.3 · V . .... 'J , - f · :}~tiq EXERCITII DE ANALIZA , REALA SI COMPLExA EDITURA PRINTECH BUCURESTI 2005
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Copyright © Printech, 2005 Editura acreditataC.N.C.S.I.S. ...!. . Descri efe'~ t' bP '! a " Bibliot~~ii ' Nafional~ ~ .. Romaniei Gheorghe OPRI~AN Exercitii de analiza reaHi ~i complexii - Gheorghe Opri~an, Antonela Toma, Crjstina Anton Bucure~ti: Printech,?-Q.oj =~~·~ :,,,, \·~ : . p.;cm. Bibliogr. ISBN 973-718-33 1-2 L) Universitatea "Politehnica" din Bucure~ti, Departamentul de matematici, Catedra de matematici II, Spl. Independentei 313, sector 6, Bucure~ti, ~ e-mail: [email protected] ~ 1 I TIPAR: I Editura PRINTECH (S.c. ANDOR TIPO S.R.L.) Str.TUNARl m.ll , sector 2 • BUCURESTr TelfFax: 211.37.12 I I © Copyright 2005 Toate drepturile prezentei editii sunt rezervate editurii si autorului. Niei 0 parte din aeeasta luerare nu poate fi reprodusa, stoeata sau transmisa indiferent prin ee forma, lara aeordul prealabil sens al autorului. -, ' . Cuprins Prefa~a .:-~p-~ ", I ENUNTURl 1 ~iruri ~i serii 1.1 $iruri numeriee 1.2 Serii numeriee . 1.3 $iruri ~i scrii de fUlletii . 1.4 Dezvoltari III serie. Dezvoltiiri limitate .. 1.5 Serii Fourier. . . . . . . . . . 2 Funetii de mai multe variabile 2.1 Limite.Continuitate. 2.2 Spa~ii metriee . 3 Calcul diferen~ial 3.1 Diferen~iabilitate.Derivatc partiale . 3.2 Extreme. FtIIlctii implieite . 4 Integrala 4.1 Primitive.lntegrale Riemann. 4.2 Teoria masurii.lntegrala Lebesgue. 4.3 Integrale improprii. .. 4.4 Integrate eu parametru . 4.5 Integrale euleriene 4.6 Integrale eurbilinii 4.7 Iutegrale multiple.Teoria campului III ." :I ,.L' V ' . 4·, 1 3 3 5 8 12 14 17 17 20 23 23 25 31 31 33 35 37 38 39 43
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4.8 Integrale de suprafa~a 4.9 Formule integrale Analiza complex8. 5.1 Fu.nc~ii olomorfe. Rela~iile Cauchy-Riemann 5.2 Dezvoltari In serie. Reziduuri ....... . I ~~-fl !: . ~~ ~ff~~::~ ~ 6 Transformata Laplace -. II IND1CATII SI RASPUNSURl L) CUPJUNS 50 52 55 55 59 65 69 Prefata Aceasta colec~ie de exerci~ii acopera cerintele cursului de Analiza. matematica care cSte predat studentilor din anul I al universitatilor tehnice. In pllll3 lucrarea contine doua capitole referitoare la teoria func~iilor de 0 variabiIa. complexa ::Ii la transformarea Laplace. Aceste elemente sunt nece- sare pentru abordarea unor integraie folosind teorema reziduurilor, precum §i pentru rezolvarea anumitor tipuri de ecuatii diferen~iale lii integrale. La problemele propuse nu s-au dat solutii ample ci doar indicatii §i raspunsnri suficiente pentru cititorul care a parcurs elementele teoretice de baza ale cursului de Analiza matematica. In acest feI rezolvitorul va capata abiliUiti in aplicarea teoriei, element esential pentru Inva~amantul tehnic. Selectarea acestor probleme reprezinta rezultatlll anilor de experienta didactica. a autorilor la faculta~ile de Electronica §i Thlecomunica~ii §i Automatica §i Calculatoare. Lucrarea este utila atat studen~ilor, cat §i cadrelor didactice din InvatiiIl.lantlll superior telmic sau cercetatorilor care doresc sa-§i amintea.<;ca despre di- versele tip uri de probleme clasice ale matematicii superioare.
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