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![[tex]D_I = \{z / |z-1| < \sqrt{2}\} , D_{II} = \{z / |z+1| < \sqrt{2}\} \ , \ D = D_I \cap D_II [/tex]](images/latex/8bb0a941c101e52c2c740fc0de047367ab33d8a9_0.png "[tex]D_I = \{z / |z-1| < \sqrt{2}\} , D_{II} = \{z / |z+1| < \sqrt{2}\} \ , \ D = D_I \cap D_II [/tex]")
Buscar una función ![[tex]U[/tex]](images/latex/9817e7675b6edb3a5d569b307744275351d202b2_0.png "[tex]U[/tex]")
armónica en ![[tex]D[/tex]](images/latex/4db8f2a83e5bb46f76520cf5386485420f2aeacf_0.png "[tex]D[/tex]")
tal que sea constante igual a 1 en la frontera de ![[tex]D_1 \in D [/tex]](images/latex/51000a08e1fc18ad3983c97ea640a34e0b99ac99_0.png "[tex]D_1 \in D [/tex]")
y constante igual a 2 en ![[tex]D_1I \in D [/tex]](images/latex/12e9f69be2418a25ec79accce55ba46913036147_0.png "[tex]D_1I \in D [/tex]")
![[tex]f(z) = \frac{(z+1)^2}{z^2+1}[/tex]](images/latex/6aa825c79ae56483bc3ee8c562858b9b59d3be50_0.png "[tex]f(z) = \frac{(z+1)^2}{z^2+1}[/tex]")
se pide el desarrollo en Series de Laurent de ![[tex]f(z)[/tex]](images/latex/c26841d08680273de3bd156265b1cbccb2108086_0.png "[tex]f(z)[/tex]")
en potencias de ![[tex]z[/tex]](images/latex/92f0f14d95ee05d626edff5beefd55475db83278_0.png "[tex]z[/tex]")
que sea convergente en ![[tex]z = 2[/tex]](images/latex/edf6380db67133405641cc7f19741c39e134c1ee_0.png "[tex]z = 2[/tex]")
![[tex]\lim_{z \to \infty} = \infty[/tex]](images/latex/b7eaf59816d130126feb3f97dc6e50c70c8c5ef3_0.png "[tex]\lim_{z \to \infty} = \infty[/tex]")
y ![[tex]\lim_{z \to 0} = \infty [/tex]](images/latex/45f35844757abfab384ac44da5ef7f1d985a079c_0.png "[tex]\lim_{z \to 0} = \infty [/tex]")
y holomorfa en ![[tex]\mathbf C - \{0\}[/tex]](images/latex/3586bf27b12b172cf1dd2a1729609c9c9f555e7c_0.png "[tex]\mathbf C - \{0\}[/tex]")
![[tex]f(z) = \frac{\pi z (1-z^2)}{sen(\pi z)}[/tex]](images/latex/878bf6e4822c06587b7ef9fd32565fd4bc19e1e7_0.png "[tex]f(z) = \frac{\pi z (1-z^2)}{sen(\pi z)}[/tex]")
Clasificar las singularidades y calcular ![[tex]Res(f,0) , Res(f, \infty)[/tex]](images/latex/57a0f60c629915a193bfac4ddcf3db408e6d873c_0.png "[tex]Res(f,0) , Res(f, \infty)[/tex]")
![[tex]\oint_\gamma \sqrt[3]{z-1} \frac{f(z)}{z} [/tex]](images/latex/32566d8d5e08842528adc27ecdec3c8d312ef0ac_0.png "[tex]\oint_\gamma \sqrt[3]{z-1} \frac{f(z)}{z} [/tex]")
donde ![[tex] \gamma : z / | z | = \frac{1}{2} [/tex]](images/latex/5f4d95240d2d5dad0433008f14fed8ee51a9fcda_0.png "[tex] \gamma : z / | z | = \frac{1}{2} [/tex]")
recorrida uina vez en sentido antihorario, utilizando la determinación de ![[tex]\sqrt[3]{z}[/tex]](images/latex/7ece6e7853e7985ff349be53daf9720761acb340_0.png "[tex]\sqrt[3]{z}[/tex]")
válida en ![[tex]\mathbf C - \{ z=iy, y>0\}[/tex]](images/latex/d2cb3374cfb993932f9e178597c94bfdeb5f17bf_0.png "[tex]\mathbf C - \{ z=iy, y>0\}[/tex]")
y con ![[tex]\sqrt[3]{8} = 2[/tex]](images/latex/472d5a3c32d4ffafb99438c1ed7c7734b9ced943_0.png "[tex]\sqrt[3]{8} = 2[/tex]")
![[tex] I = \int_{-\infty}^{+\infty} \frac{x^p}{x^{2p} + 1} sen(ax) dx[/tex]](images/latex/426a78f2ec38c87d565b69322396932c825c3389_0.png "[tex] I = \int_{-\infty}^{+\infty} \frac{x^p}{x^{2p} + 1} sen(ax) dx[/tex]")
Estudiar convergencia ![[tex]\forall p \in \mathbf R , a \in \mathbf R[/tex]](images/latex/8ac61fd23bd126908081de4820c0efde7b50daaa_0.png "[tex]\forall p \in \mathbf R , a \in \mathbf R[/tex]")
![[tex]p = -1[/tex]](images/latex/96a8466964cfabc274ba6b55bb3b04525745450d_0.png "[tex]p = -1[/tex]")
![[tex]a_n[/tex]](images/latex/84c73ddd07414ee9575e3dded97158a88d3881d0_0.png "[tex]a_n[/tex]")
tal que ![[tex]|a_n| < M r^{-n} \ \ \forall n \in \mathbf N[/tex]](images/latex/cbe4415a47e0053e8fac3bef5d79749be589007b_0.png "[tex]|a_n| < M r^{-n} \ \ \forall n \in \mathbf N[/tex]")
![[tex]\sum_{n=0}^\infty a_n z^n[/tex]](images/latex/4790c7041ae1404b398ee451daca21fbe125bc1d_0.png "[tex]\sum_{n=0}^\infty a_n z^n[/tex]")
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