AntiDemidóvich. Matemática superior. Problemas resueltos. - download pdf or read online

By AntiDemidóvich

ISBN-10: 5836004536

ISBN-13: 9785836004538

AntiDemidóvich. Matemática improved. Problemas resueltos. Variable compleja: funciones de variable compleja. T.5

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Extra info for AntiDemidóvich. Matemática superior. Problemas resueltos. Variable compleja: funciones de variable compleja. T.5

Example text

1) En Ia figura 19 se representa un arco de circunferencia unidad de angulo central igual a larg zl radianes. Es evidente que Ia longitud y de Ia cuerda que d efine el arco no supera Ia Iongitud de este. El signo de igualdad s61o es 0 X posible si arg z = 0, es decir, z E IR, z > 0. 2) Consideremos el triangulo curvilineo d e Ia figura 20. ado es Jgual a liz I - 11. El signo de igualdad s61o es posible para a rg z = 0. ,.. + Q2 +1r. 2 2 +{3, los valores de Vz2=1 tienen arg umentos a 1 + f!.. 2 y a1 + f!..

La ultima ecuaci6n tiene Ia 2 soluci6n Z = ~ lcl +it, -oo < t < +oo. Regresando a z, 2 1 t obtenemos z = - ~ + i--= (t E JR). ,.. 2 b cb 2i Im Z = -2i 1m (ad), t -ilm (ad) ac- ab ... iz2 )~·•• (Re z1KRe z2). · · . ,. Soluci6n. p2 EArgz2. f'2· Por tanto, Ia igualdad se verifica si sen lf'I sen lf'2 = 0, es decir, cuando ZJ E lR 6 z2 E JR. ,.. ,. Soluci6n. Sea lzl ~ 1. Consideremos Ia diferencia lz - al 2 - 11 - iizl 2 = (z - a)(z- a) - (1- az)(1- az) = = lz l2 - 2 2 2 iiz-az+ lal - 1 + az+az- lal 1zl = 2 = ( lal -1) (1-1zn ~ o.

2 ( El conjunto de puntos del plano C definido por Ia desigualdad 2J2 anterior es un drculo cerrado de radio - - y centro en el punto 3 1 4 z0 = - -+i-. <0111 2 2 Soluci6n. a) Seaz=x+iy;entonceslzl = v'x +y ,Rez+ 1= x + 1, y Ia igualdad analizada toma Ia forma y'x 2 + y 2 = x + 1. Elevando al cuadrado los dos miembros de Ia Ultima igualdad obtenemos y 2 = 2x + 1. Esta es Ia ecuaci6n de Ia parabola con vertice en el punto ( 1' -~, 0) y eje de simetria en Ia semirrecta = {(x, y) E~2 : x ~ -~, y = 0}.

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