By A.K. Boiarchuk... [et al.] ; traducido del ruso bajo la dirección de Viktoria O. Malishenko.
ISBN-10: 5836004544
ISBN-13: 9785836004545
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Extra info for Integración y Series
Example text
En cada una de estas regiones la integral tipo Cauchy detennma una funct6n analitica. Sea (0 E 'Y un punto arbitrario. Surge la cuesti6n sobre la existencia de los lfmites lim F(z) z-+(u zED+ y lim F(z}. z-+(o zED- ' Si estos limites existen se denominan, respectivamente, valor limite de la integral tipo Cauchy en el punto (0 por la izqtderda (derecha) de la curoa 'Y y se denotan mediante P+«o) y P-«o). Si dichos valores existieran seria muyutil establ~cer una relaci6n entre ellos y el valor principal de la integral tipo Cauchy P«o).
R ,«z -1)(z2 + 1») -1 = z+l r) r3 dz+ r2 , + 1 J' ' l)(z - . ( - -1 + -1 + -1 - -1 ) ' : :;: : O. 4 «z +1)(z2 + 1)r l)(z + i») . z-~ , ' dz+ -1 ~:::;::: Soluci6n.. Parametricemos la circunferencia , mediante z = riip, o ~ tp ~ 211". Entonces • i dz . ir dz . 1 dz = ir e 'P dtp, - .... - dtp, Idzl = r dtp = - ...... ~ , z z . y la integral I se reduce a una integral de segunda espede a 10 largo del eontomo r = (" ,or) orientado positivamente (en el sentido contrarioal de las agujas del relo;): / dz.
Sec. 4, cap. 1, t. 5), existe un punto Zo comlin a todos. los trhingulos. Dado que Zo E G*, entonces Zo ED. Como la fundon ! '(zo) (z - zo) +o:(z) (z - zo), (5) 0 cuando z - zo, es dedr, 'It € > 0 3 6(e) > 0: la(z)1 < e. Sea K 6 {z E C: Iz - Zo 1< 6}. Para valores suficientemente grandes de n E Nse tiene G~ E K 6 • Tomando en consideradon los ejemplos 1 y 2, p. 1, obtenemos donde a Jz - Zol < 6 =} J J(z) dz = = lJG;. '(Zo) (z - Zo) dz + lJG~ J = a(z) (z - zo) dz EJG:, J a(z) (z - Zo) dz, lJG;. de donde se deduce 1a estimad6n IJJ(Z)dZI < eIOG:1 2 • lJG~ Aqui 10G~1 es el perimetro del triangulo G~, ya que 10G~ SegUn el procedimiento realizado, tenemos I.
Integración y Series by A.K. Boiarchuk... [et al.] ; traducido del ruso bajo la dirección de Viktoria O. Malishenko.
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