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Notas em An´ alise Complexa Gabriel E. Pires 1998
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Conte´udo 1 Integra¸ ao 5 1.1 Teorema de Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Consequˆ encias do Teorema de Cauchy . . . . . . . . . . . . . . . . . . . . 14 1.3 ´ Indice de um Caminho Fechado . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 F´ormulas Integrais de Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 Teorema de Morera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6 Teorema de Liouville . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.7 Teorema Fundamental da ´ Algebra . . . . . . . . . . . . . . . . . . . . . . . 20 1.8 Zeros de Fun¸c˜oes Anal´ ıticas . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.9 Teorema do M´odulo M´aximo . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.10 Exerc´ ıcios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Singularidades 25 2.1 Classifica¸ c˜ao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 erie de Laurent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Exerc´ ıcios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Res´ ıduos e Aplica¸ oes 37 3.1 Teorema dos Res´ ıduos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Zeros e Polos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 alculo de Res´ ıduos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 alculo de Integrais e de S´ eries . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.1 Integrais do tipo: integraldisplay −∞ f ( x ) dx . . . . . . . . . . . . . . . . . . . . . 43 3.4.2 Integrais do tipo integraldisplay −∞ e iax f ( x ) dx . . . . . . . . . . . . . . . . . . 45 3.4.3 Integrais trigonom´ etricos . . . . . . . . . . . . . . . . . . . . . . 47 3.4.4 Valor principal de Cauchy . . . . . . . . . . . . . . . . . . . . . 48 3.4.5 Integrais de fun¸ oes multivalentes . . . . . . . . . . . . . . . . 51 3.4.6 Soma de s´ eries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.7 Exemplos diversos . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 Exerc´ ıcios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3
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4 CONTE ´ UDO
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Cap´ ıtulo 1 Integra¸ ao A uma fun¸c˜ao cont´ ınua γ : [ a,b ] C , em que [ a,b ] R ´ e um intervalo, chamamos caminho. Uma linha ´ e a imagem de um caminho, ou seja, ´ e o conjunto { γ ( t ) C : a t b } . Seja A = γ ( a ) e B = γ ( b ) . ´ E claro que a linha definida por γ ser´a percorrida do ponto A para o ponto B. Note-se que a fun¸c˜ao α ( t ) = a + ( b a ) t estabelece uma bijec¸ c˜ao entre os intervalos [0 , 1] e [ a,b ] . Seja g = γ α. Assim, a linha definida pelo caminho γ : [ a,b ] C ser´a tamb´ em definida pelo caminho g : [0 , 1] C . Podemos ent˜ao definir uma linha atrav´ es de um caminho no intervalo [0 , 1] . Seja γ : [ a,b ] C um caminho seccionalmente regular, ou seja, um caminho de classe C 1 excepto num conjunto finito de pontos do intervalo [ a,b ] (cf. [5],[2],[6]). Seja γ a respectiva imagem, S C um conjunto aberto tal que γ S e seja f : S C uma fun¸c˜ao cont´ ınua. Ent˜ao as fun¸c˜oes Re( f γ ) γ e Im( f γ ) γ ser˜ao seccionalmente cont´ ınuas no intervalo [ a,b ] e, portanto, integr´aveis em [ a,b ] . Assim, define-se integral de f ao longo do caminho γ, ou integral de f ao longo da linha γ , da forma seguinte: integraldisplay γ f ( z ) dz = integraldisplay b a f ( γ ( t )) γ ( t ) dt = integraldisplay b a Re bracketleftbig f ( γ ( t )) γ ( t ) bracketrightbig dt + i integraldisplay b a Im bracketleftbig f ( γ ( t )) γ ( t ) bracketrightbig dt. (1.0.1) Lema 1.0.1 Seja γ : [ a,b ] C um caminho seccionalmente regular, S C um conjunto aberto tal que γ S e f : S C uma fun¸ ao cont´ ınua. 1. integraltext γ f ( z ) dz = integraltext γ f ( z ) dz. 5
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6 CAP ´ ITULO 1. INTEGRAC ¸ ˜ AO 2. Seja ψ : [ α,β ] [ a,b ] uma fun¸ ao de classe C 1 com derivada positiva e seja ˜ γ = γ ψ uma reparametriza¸c˜ ao. Ent˜ ao, integraltext ˜ γ f ( z ) dz = integraltext γ f ( z ) dz.
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