{"id":54,"date":"2017-09-04T13:53:41","date_gmt":"2017-09-04T16:53:41","guid":{"rendered":"http:\/\/www.professores.uff.br\/jcolombo\/?page_id=54"},"modified":"2018-03-13T12:09:52","modified_gmt":"2018-03-13T15:09:52","slug":"variaveis-complexas-gan00169","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/jcolombo\/variaveis-complexas-gan00169\/","title":{"rendered":"Vari\u00e1veis Complexas – GAN00169 – 2\/2017"},"content":{"rendered":"

Carga hor\u00e1ria:<\/strong> 68 hs, turma A1<\/p>\n

\n

Monitoria: <\/strong>n\u00e3o temos monitor.
\n<\/strong><\/p>\n

\n

Lista de Exerc\u00edcios para a P1<\/strong>
\nLista 1:\u00a0 Exerc\u00edcios das p\u00e1ginas 5-6, 12-13, 16-18 e 21 do livro do Churchill, vers\u00e3o inglesa.
\nLista 2:\u00a0 Exerc\u00edcios das p\u00e1ginas 32-33, 38-39, 44-45\u00a0 do livro do Churchill, vers\u00e3o inglesa.
\nLista 3:\u00a0 Exerc\u00edcios das p\u00e1ginas 55-56, 59, 61, 67, 71\u00a0 do livro do Churchill, vers\u00e3o inglesa.
\nLista 4:\u00a0 Exerc\u00edcios das p\u00e1ginas 77-78, 83-85, 94-95, 100, 103-104, 107, 110\u00a0 do livro do Churchill, vers\u00e3o inglesa.
\nLista 5:\u00a0 Exerc\u00edcios das p\u00e1ginas 141-142, 153-156, 162-164 e 171-173 do livro do Churchill, vers\u00e3o inglesa.<\/p>\n

Lista de Exerc\u00edcios para a P2<\/strong><\/p>\n

Lista 5: \u00a0Exerc\u00edcios das p\u00e1ginas \u00a0181-183, 188-190, 198-200 do livro do Churchill, vers\u00e3o inglesa.
\nLista 6: \u00a0Exerc\u00edcios das p\u00e1ginas \u00a0212-215, 218-220, 240 do livro do Churchill, vers\u00e3o inglesa.
\nLista 7: \u00a0Exerc\u00edcios das p\u00e1ginas \u00a0257-259, 265-267, 276-278, 280, 285-288 do livro do Churchill, vers\u00e3o inglesa.
\nLista 8: \u00a0Exerc\u00edcios das p\u00e1ginas \u00a0316-318 do livro do Churchill, vers\u00e3o inglesa.
\nLista 9: \u00a0Exerc\u00edcios das p\u00e1ginas: \u00a078, 80, 82-83 e 88 da refer\u00eancia -1+i) Ahlfors<\/p>\n

\n

Hor\u00e1rio:<\/strong> 3\u00aa e 5\u00aa-feiras na sala IM-401H das 9 \u00e0s 11hs.
\nMeu hor\u00e1rio de atendimento: <\/strong>4\u00aa-feiras das 14-16hs.
\n<\/strong><\/p>\n

\n

Provas
\n<\/strong><\/p>\n

\n
P1<\/a> – 10\/10\/2017 – Gabarito feito em aula<\/dt>\n
P2<\/a> – 07\/12\/2017 – Gabarito da P2<\/a><\/dt>\n<\/dl>\n
\n
VR<\/a> -17\/12\/2017 –<\/dt>\n
VS -19\/12\/2017 (esta avalia\u00e7\u00e3o n\u00e3o foi necess\u00e1ria)<\/dt>\n<\/dl>\n
\n

Cronograma<\/strong><\/p>\n

Aula \u00a0 \u00a0 Data\u00a0\u00a0\u00a0 Assunto
\n01\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 22\/08 \u00a0\u00a0Apresenta\u00e7\u00e3o \u00a0da \u00a0disciplina, ementa, bibliografia e avalia\u00e7\u00e3o, defini\u00e7\u00e3o de n\u00famero complexo e propriedades.
\n02\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 24\/08 \u00a0\u00a0Desigualdades, ra\u00edzes n-\u00e9simas da unidade, fun\u00e7\u00e3o exponencial..
\n03\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 29\/08 \u00a0\u00a0Defini\u00e7\u00f5es b\u00e1sicas de topologia do plano Complexo, fun\u00e7\u00f5es complexas, gr\u00e1fico, limite, continuidade.
\n04\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 31\/08 \u00a0\u00a0Fun\u00e7\u00f5es An\u00e1liticas, equa\u00e7\u00f5es de Cauchy-Riemann, fun\u00e7\u00f5es trignom\u00e9tricas e hiperb\u00f3licas.
\n05 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a005\/09 \u00a0 Exponencial, \u00a0logaritmo, superf\u00edcie de Rieman,\u00a0 z^{\\alpha}.
\n07\/09 \u00a0\u00a0Feriado – Dia da Independ\u00eancia.
\n06 \u00a0\u00a0 \u00a0\u00a0\u00a0 12\/09\u00a0 \u00a0Fun\u00e7\u00f5es trignom\u00e9tircas inversas, fun\u00e7\u00f5es harm\u00f4nicas. Interpreta\u00e7\u00e3o de fun\u00e7\u00f5es complexas como aplica\u00e7\u00f5es ou transforma\u00e7\u00f5es do plano.
\n07 \u00a0\u00a0 \u00a0\u00a0\u00a0 14\/09 \u00a0\u00a0Representa\u00e7\u00e3o esf\u00e9rica. Integral: Defini\u00e7\u00e3o de caminhos, contornos, integral de contorno
\n08 \u00a0\u00a0 \u00a0\u00a0\u00a0 19\/09 \u00a0 Integral complexa: teorema de Cauchy
\n09 \u00a0\u00a0 \u00a0\u00a0\u00a0 21\/09 \u00a0 Coment\u00e1rios sobre o\u00a0teorema de Cauchy-Goursat, integral em regi\u00f5es multiplas conexas, Transforma\u00e7\u00f5es de M\u00f6bius e raz\u00e3o cruzada
\n10 \u00a0\u00a0 \u00a0\u00a0\u00a0 26\/09 \u00a0\u00a0F\u00f3rmula Integral de Cauchy, Derivadas de Ordem Superior e Teorema de Morera.
\n11 \u00a0\u00a0 \u00a0\u00a0\u00a0 28\/09 \u00a0 Fun\u00e7\u00f5es harm\u00f4nicas. Teorema do m\u00f3dulo m\u00e1ximo
\n12 \u00a0\u00a0 \u00a0\u00a0\u00a0 03\/10 \u00a0 Revimos o teorema do m\u00f3dulo m\u00e1ximo e exerc\u00edcios
\n13 \u00a0\u00a0 \u00a0\u00a0\u00a0 05\/10 \u00a0\u00a0Aula de exerc\u00edcios
\n14 \u00a0\u00a0 \u00a0\u00a0\u00a0 10\/10 \u00a0\u00a0P1<\/b>
\n12\/10 \u00a0 Feriado – \u00a0gabarito on-line
\n15 \u00a0\u00a0 \u00a0\u00a0\u00a0 17\/10 \u00a0 S\u00e9ries: Converg\u00eancia,\u00a0converg\u00eancia absoluta e s\u00e9ries de Taylor;
\n16 \u00a0\u00a0 \u00a0\u00a0\u00a0 19\/10 \u00a0 S\u00e9ries\u00a0de Laurent.
\n17 \u00a0\u00a0 \u00a0\u00a0\u00a0 24\/10 \u00a0 Semana Acad\u00eamica
\n18 \u00a0\u00a0 \u00a0\u00a0\u00a0 26\/10 \u00a0 Semana Acad\u00eamica
\n19 \u00a0\u00a0 \u00a0\u00a0\u00a0 31\/10 \u00a0 Singulariedades, polos, zeros, res\u00edduos e integrais.
\n02\/11 \u00a0 Feriado
\n20 \u00a0\u00a0 \u00a0\u00a0\u00a0 07\/11 \u00a0\u00a0Aula de exerc\u00edcios – S\u00e9ris de Laurent
\n21 \u00a0\u00a0 \u00a0\u00a0\u00a0 09\/11 \u00a0 Aula de exerc\u00edcios –\u00a0 \u00a0Polos, zeros e Res\u00edduos
\n22 \u00a0\u00a0 \u00a0\u00a0\u00a0 14\/11 \u00a0 Aplica\u00e7\u00f5es: Avaliando integrais Improprias
\n23\u00a0 \u00a0\u00a0 \u00a0\u00a0\u00a016\/11 \u00a0 Aplica\u00e7\u00f5es: Integra\u00e7\u00e3o ao redor de um ponto de ramifica\u00e7\u00e3o.
\n24 \u00a0\u00a0 \u00a0\u00a0\u00a0 21\/11 \u00a0 Feriado
\n26 \u00a0\u00a0 \u00a0\u00a0\u00a0 23\/11 \u00a0 Aplica\u00e7\u00f5es: Teorema de Rouch\u00e8
\n27 \u00a0\u00a0 \u00a0\u00a0\u00a0 28\/11 \u00a0 + propriedades da transforma\u00e7\u00f5es de M\u00f6bius: Invers\u00e3o, invari\u00e2n\u00e7a da raz\u00e3o cruzada, ponto fixo.
\n28 \u00a0\u00a0 \u00a0\u00a0\u00a0 30\/11 \u00a0 Classifica\u00e7\u00e3o das transforma\u00e7\u00f5es de M\u00f6bius, aula de exerc\u00edcios.
\n29 \u00a0\u00a0 \u00a0\u00a0\u00a0 05\/12 \u00a0 Aula de exerc\u00edcios
\n30 \u00a0\u00a0 \u00a0\u00a0\u00a0 07\/12 \u00a0 P2
\n31 \u00a0\u00a0 \u00a0\u00a0\u00a0 12\/12 \u00a0 Vista da P2
\n32 \u00a0\u00a0 \u00a0\u00a0\u00a0 14\/12 \u00a0 VR<\/b>
\n33 \u00a0\u00a0 \u00a0\u00a0\u00a0 19\/12 \u00a0 VS<\/b><\/p>\n

\n

Bibliografia:<\/strong><\/p>\n

1) Ruel V. Churchill, Complex Variables and Applications, 3nd edition, McGraw-Hill.
\n1+i) Geraldo G. S. \u00c1vila, Vari\u00e1veis Complexas e Aplica\u00e7\u00f5es, LTC, 1990.
\ni) \u00a0Mario G. Souza, Calculo em uma vari\u00e1vel complexa, CMU, IMPA, 2001.
\n-1+i) Lars V. Ahlfors, Complex Analysis, McGraw-Hill International company, 1979.
\n-1) Cec\u00edlia. S. Fernandez e Nilson C. Bernardes, Introdu\u00e7\u00e3o \u00e0s Fun\u00e7\u00f5es de uma Vari\u00e1vel Complexa, SBM.<\/p>\n

\n

Ementa:<\/strong><\/p>\n

1 – N\u00fameros Complexos e o plano complexo. Forma polar. Teorema de De Moivre e extra\u00e7\u00e3o de ra\u00edzes.
\n2 – Topologia do Plano Complexo: Conjuntos abertos, fechados, compactos, conexos e simplesmente conexos; pontos de acumula\u00e7\u00e3o e pontos de fronteira.
\n3 – Fun\u00e7\u00f5es complexas: exemplos, limite de fun\u00e7\u00e3o num ponto, opera\u00e7\u00f5es com limites; fun\u00e7\u00f5es cont\u00ednuas.
\n4 – Fun\u00e7\u00f5es elementares: exponencial, logaritmo, trigonom\u00e9trica.
\n5 – Fun\u00e7\u00f5es anal\u00edticas; equa\u00e7\u00f5es de Cauchy-Riemann; ramos de fun\u00e7\u00f5es anal\u00edticas; fun\u00e7\u00f5es harm\u00f4nicas.
\n6 – Integral de fun\u00e7\u00f5es complexas; Teorema de Jordan; Teorema de Cauchy; Teorema de Cauchy-Goursat (quadrado, tri\u00e2ngulo ou disco).
\n7. F\u00f3rmula Integral de Cauchy e aplica\u00e7\u00f5es.
\n8 – Zero de ordem m de uma fun\u00e7\u00e3o anal\u00edtica; Teorema do M\u00f3dulo M\u00e1ximo.
\n9 -Sequ\u00eancias e s\u00e9ries de n\u00fameros complexos; s\u00e9ries de potencias; converg\u00eancia uniforme; s\u00e9ries de Taylor e de Laurent.
\n10 – Singularidades isolada, essencial e remov\u00edvel.
\n11 – Polos e Res\u00edduos; Teorema dos Res\u00edduos.
\n12 – Aplica\u00e7\u00f5es do teorema dos res\u00edduos: integrais reais impr\u00f3prias, integrais definidas trigonom\u00e9tricas, integrais impr\u00f3prias de fun\u00e7\u00f5es complexas e integra\u00e7\u00e3o envolvendo um ponto de ramifica\u00e7\u00e3o.<\/p>\n

\n

\u00daltima revis\u00e3o em Dezembro, 2017.<\/p>\n","protected":false},"excerpt":{"rendered":"

Carga hor\u00e1ria: 68 hs, turma A1 Monitoria: n\u00e3o temos monitor. Lista de Exerc\u00edcios para a P1 Lista 1:\u00a0 Exerc\u00edcios das p\u00e1ginas 5-6, 12-13, 16-18 e 21 do livro do Churchill, vers\u00e3o inglesa. Lista 2:\u00a0 Exerc\u00edcios das p\u00e1ginas 32-33, 38-39, 44-45\u00a0 do livro do Churchill, vers\u00e3o inglesa. Lista 3:\u00a0 Exerc\u00edcios das p\u00e1ginas 55-56, 59, 61, 67, […]<\/p>\n","protected":false},"author":70,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[],"tags":[],"class_list":["post-54","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/pages\/54","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/comments?post=54"}],"version-history":[{"count":10,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/pages\/54\/revisions"}],"predecessor-version":[{"id":674,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/pages\/54\/revisions\/674"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/media?parent=54"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/categories?post=54"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/jcolombo\/wp-json\/wp\/v2\/tags?post=54"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}