MATEMATICĂ 3 PROBLEME DE REFLECŢIE

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1 Recapitulare din liceu MATEMATIĂ 3 ANALIZĂ OMPLEXĂ PROBLEME DE REFLEŢIE. Scrieţi numerele următoare sub forma a + bi, unde a, b R: a) 3i + i ; b) i + i ;. Reolvaţi în ecuaţiile: ( + i)( i) c) ( + i)(4 3i) ; (4 + 5i) + i3 d). ( + i) a) + 7 6i = ; b) + = i + 3i. 3. Descrieţi mulţimea punctelor din planul complex, care verifică relaţiile: a) Re(( + i) ) = ; b) = i ; c) = ; d) Re( ) = 3 i ; e) i = ; f) =Re(). 4. a) Dacă =, arătaţi că 3 4. b) Găsiţi o majorare convenabilă pentru 3 + +, dacă. 5. Găsiţi un număr complex ce verifică ecuaţiile: a) = + i; b) + + i = Pentru orice două numere complexe şi, arătaţi că: + + = ( + ). 7. Scrieţi următoarele numere complexe în formă polară: a) 3i; b) 5 5i; c) 3i; d). 3 + i ( 8. alculaţi + ) i. 9. Arătaţi că + cos θ + cos θ cos nθ = + sin ( ) n + θ sin θ, unde < θ < π.. Reolvaţi ecuaţia: = ( + i) 5.. Repreentaţi grafic soluţiile următoarelor relaţii:

2 a) + 3i = ; b) = 4; c) Re( )=; d) <Re( ) < 4; e) i < ; f) 3 + 4i 5. Funcţii complexe. Evaluaţi funcţia f() = ln + iarg() în punctele: a) ; b) 4i; c) + i. 3. Aflaţi părţile reale şi imaginare (u şi v) ale funcţiilor: a) f() = 3 + 6; b) f() = + ; c) f() = + ; d) f() = e +i. 4. Găsiţi domeniul şi imaginea funcţiei f() = Arătaţi că funcţia f() = ei + e i este periodică de perioadă π. 6. Găsiţi imaginea S a mulţimii S, prin funcţia complexă w = f(), dacă: (a) f() =, S este dreapta y = x; (b) f() = 3, S este semiplanul cu Im() > ; (c) f() = 3, S este banda verticală infinită Re() < 3; (d) f() = i + 4, S este banda oriontală infinită < Im() <. Limită şi continuitate 7. alculaţi: (a) lim +i ( + i); (b) lim i e. 8. alculaţi: (a) lim ( ); i (b) lim 5 + ; i (c) (d) lim e iπ/4 lim +i ( + + ; ) ; 4 (e) lim i + i ; ( + i) (f) lim +i ( + i) ; (g) lim + i ( + i) ; (h) lim i + i ; (i) lim i + ; (j) lim i i. 9. Arătaţi că f este continuă în punctul indicat: (a) f() = i + 3 i, = i; (b) f() = Re() + i, = e iπ/4 ;

3 3, = (c) f() = i, 3 = + i 3., = Derivabilitate. Olomorfie. Folosind proprietăţile derivării, calculaţi f () pentru funcţiile: a) f() = ( i) 5 +i 4 3 +i 6 ; b) f() = ( 6 )( + 5i); c) f() = i 3 + i ; d) f() = ( 4 i + )., =. Arătaţi că funcţia f() = x y 3 x + y + + y 3 ix3 x + y, renţiabilă în =. nu este dife-. Utiliaţi regula lui Guillaume de L Hôpital pentru a calcula limitele: 7 + i a) lim i 4 + ; b) lim +i + ; c) lim + i ; d) lim i Arătaţi că următoarele funcţii nu sunt analitice: a) f() =Re(); b) f() = ; c) f() = ; x d) f() = x + y + i y x + y. 4. Arătaţi că următoarele funcţii sunt analitice pe un domeniu adecvat, folosind ecuaţiile auchy -Riemann şi continuitatea derivatelor parţiale ale funcţiilor u şi v: (a) f() = e x cos y ie x sin y; (b) f() = 4x + 5x 4y i(8xy + 5y ); x (c) f() = (x ) + y i y (x ) + y ; (d) f() = cos θ i sin θ r r. 5. Aflaţi numerele a, b, c, d, pentru care următoarele funcţii sunt analitice: (a) f() = 3x y i(ax + by 3); (b) f() = x + axy + by + i(cx + dxy + y ). Augustin auchy ( ) a fost unul dintre cei mai importanţi matematicieni francei. A fost unul dintre pionierii analiei matematice şi a adus o serie de contribuţii şi în domeniul fiicii. Bernhard Riemann (86-866) a fost un matematician german cu importante contribuţii în analia matematică şi geometria diferenţială, unele dintre ele deschiând drumul ulterior spre teoria relativităţii generaliate. 3

4 6. Arătaţi că următoarele funcţii nu sunt analitice, dar sunt diferenţiabile de-a lungul curbelor indicate: (a) f() = x + y + ixy, axa Ox; (b) f() = 3x y 6ix y, axele de coordonate; (c) f() = x x + y + i(y 5y x), dreapta y = x Dacă f() este o funcţie analitică pe un domeniu D şi f() = c, unde c este o constantă reală, arătaţi că funcţia f() este constantă pe D. 8. Arătaţi că următoarele funcţii sunt armonice pe un domeniu adecvat, găsiţile conjugatele armonice şi funcţiile f() din care provin: a) u(x, y) = xy + x + y, f(i) = + 5i; b) u(x, y) = 4xy 3 4x 3 y + x, f( + i) = 5 + 4i. Funcţii elementare 9. alculaţi derivatele următoarelor funcţii: a) f() = e +i ; c) f() = e i e i ; b) f() = 3e ie 3 + i ; d) f() = ie. 3. Aduceţi funcţiile f la forma f() = u(x, y) + iv(x, y): a) f() = e i ; c) f() = e ; b) f() = e +i ; d) f() = e. 3. Găsiţi imaginea benii infinite < x prin transformarea w = e. 3. Aflaţi valorile logaritmilor complecşi: a) ln( 5); d) ln( 3 + i); b) ln( ei); e) ln( + i); c) ln( + i); f) Ln(6 6i); g) Ln( e ); h) Ln [ ( + 3i) 5] ; i) Ln[( + i) 4 ]. 33. Reolvaţi ecuaţiile: a) e = ie 3 ; b) e + e + = ; c) e =. 34. Determinaţi domeniul în care funcţia f este diferenţiabilă şi calculaţi derivata f : a) f() = 3 e i + iln; Ln( i) c) f() = ; + b) f() = ( + )Ln( + ); d) f() =Ln( + ). 35. Determinaţi valorile principale ale următoarelor puteri complexe: a) ( ) 3i ; c) i i π ; b) 3 i π ; d) ( + i) i. 36. Exprimaţi valorile funcţiilor de mai jos sub forma x + iy: a) sin(4i); b) cos( 4i); c) tan(i); d) csc( + i); e) cosh(πi); f) tanh( + 3i). 37. Reolvaţi ecuaţiile: 4

5 a) sin = i; b) cos = 4; c) cosh = i; d) sinh = e. 38. alculaţi derivatele următoarelor ( ) funcţii f(): a) sin( ); c) tan ; b) cos(ie ); d) sec( + ( i)); e) sin sinh ; f) cosh(i + e i ). 39. Determinaţi valorile: a) cos i; b) sin ; c) tan (i); d) tanh ( + i). 4. Reolvaţi ecuaţiile de gradul al doilea: a) + i = ; c) + 3i = ; b) i + i = ; d) 3 + ( 3i) 3i =. Integrale complexe 4. Evaluaţi următoarele integrale de-a lungul curbei indicate: a) ( ) d, unde este: x = t, y = t +, t ; b) d, unde este: x = t, y = t, t < ; ( c) ( + i) 5 ) 3 + i + 8 d, unde este cercul + i = ; d) sin d, unde este linia poligonală formată din segmentele ce unesc = cu = şi = cu = + i. 4. Găsiţi o majorare convenabilă pentru valorile absolute ale integralelor următoare, de-a lungul curbei indicate: e a) d, unde este cercul = 5; + b) ( + 4) d, unde este segmentul ce uneşte = cu = + i; c) d, unde este sfertul cercului = 4, de la = 4i la = 4. 3 Teorema auchy-goursat Evaluaţi următoarele integrale de-a lungul curbei indicate: ( a) ( 3 + 3i) d, = ; b) + ) d, = ; 4 3 Edouard Goursat ( ) a fost un matematician france, cunoscut în special pentru lucrarea ours d analyse mathématique, care a fixat un standard înalt pentru analia matematică, în special pentru analia complexă. 5

6 c) d, = ; 9 g) d, = ; + 3 ( cosh 3 d) d, = ; h) + ) d, = ; + + sin e) d, = ; ( i) d, = 3; 5) e π f) d, = ; j) d, + i = ; ( + i) 4 + k) + d, (i) =, (ii) =, (iii) 3i = ; 3 + l) 8 + d, (i) 5 =, (ii) i = ; ( 3 m) + ) d, (i) = 5, (ii) i = i ; n) ( i)( 3i) d, i = ; ( ) e o) d, = ; ( p) Re() ) d, unde este triunghiul de vârfuri: =, = + i şi =. Formulele lui auchy 44. Dacă = { = }, calculaţi: e ) i d; 4) e ) i d; 5) e 3) + d; 6) Teorema reiduurilor e 5 ( i) 3 d; e ( + ) d; 4 ( 5) d. 45. Evaluaţi următoarele integrale reale, utiliând teorema reiduurilor: ) ) 3) x + dx; x 4 + dx; x 6 + dx; 4) 5) 6) (x + ) dx; (x + ) 3 dx; (x + )(x + 4) dx; 6

7 7) 8) 9) ) ) ) 3) 4) 5) 6) π π π π π π x + x (x + )(x + 4) dx; x x 4 + dx; x x 4 + 5x + 4 dx; (4x + ) 3 dx; a + cos θ cos 3t 5 4 cos t dt; dθ, a > ; sin θ dθ; ( + cos θ) dθ; + a sin θ + a cos θ dθ, < a < ; dθ, < a < ; 7) 8) 9) ) ) ) 3) π < p < ; 4) 5) 6) π x sin x x + 9 dx; sin x x dx; sin x x + x + dx; cos x dx, a > ; x + a cos mx dx, m > ; a + x cos mx dx, m > ; (a + x ) p cos θ + p dθ, (cos 3 t + sin t) dt; x x + dx; dx. (x + ) n+ Bibliografie [] Mary L. Boas, Mathematical Methods in the Physical Sciences, Wiley, 5. [] Murray Spiegel, Seymour Lipschut, John Schiller, Dennis Spellman, Schaum s Outline of omplex Variables, McGraw-Hill, 9. [3] Dennis G. Zill, Patrick Shanahan, A First ourse in omplex Analysis with Applications, Jones & Bartlett Learning, 8. [4] Alexandru Negrescu Universitatea Politehnica din Bucureşti 7

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