6 files changed, 4613 insertions, 0 deletions

diff --git a/README.md b/README.md
index c1b2482..2f2da39 100644
--- a/README.md
+++ b/README.md
@@ -21,6 +21,8 @@ son las siguientes:
 * `fvv3`: Funciones de Varias Variables III (sólo PDF).
 * `edo`: Ecuaciones Diferenciales Ordinarias (sólo PDF).
 * `epe`: Elementos de Probabilidad y Estadística (sólo PDF).
+* `cn`: Cálculo Numérico (sólo PDF).
+* `fvc`: Funciones de Variable Compleja (falta la primera parte).
 
 Las asignaturas de ingeniería informática que se incluyen son las siguientes:
 
diff --git a/cn/n.pdf b/cn/n.pdf
new file mode 100644
index 0000000..87a0122
--- /dev/null
+++ b/cn/n.pdf
diff --git a/fvc/n.lyx b/fvc/n.lyx
new file mode 100644
index 0000000..9d4873d
--- /dev/null
+++ b/fvc/n.lyx
@@ -0,0 +1,181 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
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+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize 10
+\spacing single
+\use_hyperref false
+\papersize a5paper
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 0.2cm
+\topmargin 0.7cm
+\rightmargin 0.2cm
+\bottommargin 0.7cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle empty
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Funciones de variable compleja
+\end_layout
+
+\begin_layout Date
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+def
+\backslash
+cryear{2020}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "../license.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Bibliografía:
+\end_layout
+
+\begin_layout Itemize
+Curso de Análisis Complejo, Francisco Javier Pérez González (2004), Departamento
+ de Análisis Matemático, Universidad de Granada.
+\end_layout
+
+\begin_layout Chapter
+Teoría de Cauchy elemental
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n2.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Ceros de funciones holomorfas
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n3.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Forma general del teorema de Cauchy
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n4.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvc/n2.lyx b/fvc/n2.lyx
new file mode 100644
index 0000000..b18a007
--- /dev/null
+++ b/fvc/n2.lyx
@@ -0,0 +1,1670 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Teorema de Cauchy en dominios estrellados
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Cauchy-Goursat:
+\series default
+ Sea 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $\Delta(a,b,c):=\{\mu a+\lambda b+\gamma c:\mu+\lambda+\gamma=1;\mu,\lambda,\gamma\geq0\}\subseteq\Omega$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\int_{[a,b,c,a]}f=0.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $\gamma:=[a,b,c,a]$
+\end_inset
+
+, 
+\begin_inset Formula $\Delta:=\Delta(a,b,c)$
+\end_inset
+
+, 
+\begin_inset Formula $a':=\frac{b+c}{2}$
+\end_inset
+
+, 
+\begin_inset Formula $b':=\frac{a+c}{2}$
+\end_inset
+
+, 
+\begin_inset Formula $c':=\frac{a+b}{2}$
+\end_inset
+
+ e
+\begin_inset Formula 
+\[
+I:=\int_{\gamma}f=\int_{[a,c',b',a]}f+\int_{[c',b,a',c']}f+\int_{[a',c,b',a']}f+\int_{[b',c',a',b']}f.
+\]
+
+\end_inset
+
+Sean 
+\begin_inset Formula $J_{1},\dots,J_{4}$
+\end_inset
+
+ las cuatro integrales a la derecha, 
+\begin_inset Formula $\sigma_{1},\dots,\sigma_{4}$
+\end_inset
+
+ las correspondientes curvas y 
+\begin_inset Formula $T_{1},\dots,T_{4}$
+\end_inset
+
+ los triángulos definidos por estas curvas.
+ Entonces:
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $|J_{k}|:=\max_{i}|J_{i}|$
+\end_inset
+
+, 
+\begin_inset Formula $|I|\leq4|J_{k}|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\ell(\sigma_{1})=\dots=\ell(\sigma_{4})=\frac{1}{2}\ell(\gamma)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\ell(\sigma_{1})=|a-c'|+|c'-b'|+|b'-a|=\left|a-\frac{a+b}{2}\right|+\left|\frac{a+b}{2}-\frac{a+c}{2}\right|+\left|\frac{a+c}{2}-a\right|=\left|\frac{a-b}{2}\right|+\left|\frac{b-c}{2}\right|+\left|\frac{c-a}{2}\right|=\frac{1}{2}(|a-b|+|b-c|+|c-a|)=\frac{1}{2}\ell(\gamma)$
+\end_inset
+
+.
+ Para el resto de curvas se hace algo análogo.
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+Sea 
+\begin_inset Formula $d.(S)$
+\end_inset
+
+ el diámetro de 
+\begin_inset Formula $S\subseteq\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $D(T_{1})=\dots=D(T_{4})=\frac{1}{2}D(\Delta)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $T_{1}$
+\end_inset
+
+, 
+\begin_inset Formula $F(x):=\frac{x+a}{2}$
+\end_inset
+
+ es una biyección de 
+\begin_inset Formula $\Delta$
+\end_inset
+
+ a 
+\begin_inset Formula $T_{1}$
+\end_inset
+
+, pues si 
+\begin_inset Formula $x:=ra+sb+tc$
+\end_inset
+
+, 
+\begin_inset Formula $F(x):=\frac{ra+sb+tc+a}{2}=\frac{ra+sb+tc+(r+s+t)a}{2}=ra+s\frac{a+b}{2}+t\frac{a+c}{2}=ra+sc'+tb'$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $D(T_{1})=\sup_{x,y\in T_{1}}|x-y|=\sup_{x,y\in\Delta}|F(x)-F(y)|=\sup_{x,y\in\Delta}\left|\frac{x+a}{2}-\frac{y+a}{2}\right|=\sup_{x,y\in\Delta}\frac{|x+y|}{2}=\frac{1}{2}D(\Delta)$
+\end_inset
+
+.
+ Para los otros triángulos se hace de forma análoga, usando para 
+\begin_inset Formula $T_{4}$
+\end_inset
+
+ la biyección 
+\begin_inset Formula $F(x):=\frac{a+b+c-x}{2}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Sean entonces 
+\begin_inset Formula $I_{1}:=\max_{i}|J_{i}|$
+\end_inset
+
+, 
+\begin_inset Formula $\gamma_{1}:=[a_{1},b_{1},c_{1},a_{1}]$
+\end_inset
+
+ la curva correspondiente a 
+\begin_inset Formula $I_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Delta_{1}:=\Delta(a_{1},b_{1},c_{1})$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $|I|\leq4|I_{1}|$
+\end_inset
+
+, 
+\begin_inset Formula $\ell(\gamma_{1})=\frac{1}{2}\ell(\gamma)$
+\end_inset
+
+ y 
+\begin_inset Formula $D(\Delta_{1})=\frac{1}{2}\Delta$
+\end_inset
+
+.
+ Repitiendo este proceso se obtienen sucesiones donde 
+\begin_inset Formula $|I|\leq4^{n}|I_{n}|$
+\end_inset
+
+, 
+\begin_inset Formula $\ell(\gamma_{n})=\frac{1}{2^{n}}\ell(\gamma)$
+\end_inset
+
+ y 
+\begin_inset Formula $D(\Delta_{n})=\frac{1}{2^{n}}D(\Delta)$
+\end_inset
+
+.
+ Al ser 
+\begin_inset Formula $(\Delta_{n})_{n}$
+\end_inset
+
+ una sucesión decreciente de cerrados no vacíos donde el diámetro tiende
+ a 0, existe un único 
+\begin_inset Formula $\alpha\in\bigcap_{n}\Delta_{n}$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $p(z):=f(\alpha)+f'(\alpha)(z-\alpha)$
+\end_inset
+
+ una función polinómica y por tanto con primitiva, entonces
+\begin_inset Formula 
+\[
+I_{n}=\int_{\gamma_{n}}f=\int_{\gamma_{n}}f-\int_{\gamma_{n}}p=\int_{\gamma_{n}}(f(z)-f(\alpha)-f'(\alpha)(z-\alpha))dz.
+\]
+
+\end_inset
+
+Dado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, como 
+\begin_inset Formula $f$
+\end_inset
+
+ es derivable en 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ existe 
+\begin_inset Formula $\delta>0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $D(\alpha,\delta)\subseteq\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $\forall z\in D(\alpha,\delta),|f(z)-f(\alpha)-f'(\alpha)(z-\alpha)|\leq\varepsilon|z-\alpha|$
+\end_inset
+
+.
+ Dado 
+\begin_inset Formula $n$
+\end_inset
+
+ con 
+\begin_inset Formula $D(\Delta_{n})<\delta$
+\end_inset
+
+, 
+\begin_inset Formula $\Delta_{n}\subseteq D(\alpha,\delta)$
+\end_inset
+
+ y
+\begin_inset Formula 
+\begin{multline*}
+|I|\leq4^{n}|I_{n}|\leq4^{n}\ell(\gamma_{n})\max_{z\in\gamma_{n}^{*}}|f(z)-f(\alpha)+f'(\alpha)(z-\alpha)|\leq4^{n}\ell(\gamma_{n})\varepsilon\max_{z\in\gamma_{n}^{*}}|z-\alpha|\leq\\
+\leq4^{n}\ell(\gamma_{n})\varepsilon D(\Delta_{n})=4^{n}\varepsilon\frac{1}{2^{n}}\ell(\gamma)\frac{1}{2^{n}}D(\Delta)=\varepsilon\ell(\gamma)D(\Delta),
+\end{multline*}
+
+\end_inset
+
+y haciendo tender 
+\begin_inset Formula $\varepsilon\to0$
+\end_inset
+
+ se obtiene el resultado.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Cauchy para dominios estrellados:
+\series default
+ Sea 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ un dominio estrellado en 
+\begin_inset Formula $z_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+F(z):=\int_{[z_{0},z]}f
+\]
+
+\end_inset
+
+es una primitiva de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $\rho>0$
+\end_inset
+
+ y 
+\begin_inset Formula $z\in D(a,\rho)$
+\end_inset
+
+, como 
+\begin_inset Formula $[a,z]\subseteq\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $[z_{0},b]\subseteq\Omega$
+\end_inset
+
+ para todo 
+\begin_inset Formula $b\in[a,z]$
+\end_inset
+
+ y 
+\begin_inset Formula $\Delta(z_{0},a,z)\subseteq\Omega$
+\end_inset
+
+.
+ Por el teorema de Cauchy-Goursat,
+\begin_inset Formula 
+\[
+0=\int_{[z_{0},z,a,z_{0}]}f=\int_{[z_{0},z]}f+\int_{[z,a]}f+\int_{[a,z_{0}]}f=F(z)-F(a)-\int_{[z,a]}f,
+\]
+
+\end_inset
+
+luego si 
+\begin_inset Formula $z\neq a$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\frac{F(z)-F(a)-f(a)(z-a)}{z-a}=\frac{\int_{[z,a]}f-f(a)(z-a)}{z-a}=\frac{\int_{[z,a]}(f(w)-f(a))dw}{z-a},
+\]
+
+\end_inset
+
+con lo que
+\begin_inset Formula 
+\[
+\left|\frac{F(z)-F(a)}{z-a}-f(a)\right|=\left|\frac{\int_{[z,a]}(f(w)-f(a))dw}{z-a}\right|\leq\max_{w\in[a,z]^{*}}|f(w)-f(a)|.
+\]
+
+\end_inset
+
+Como 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $a$
+\end_inset
+
+, haciendo 
+\begin_inset Formula $z\to a$
+\end_inset
+
+ este máximo tiende a 0 y se obtiene 
+\begin_inset Formula $F'(a)=f(a)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Cauchy-Goursat 
+\begin_inset Quotes cld
+\end_inset
+
+light
+\begin_inset Quotes crd
+\end_inset
+
+:
+\series default
+ Sean 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ un abierto, 
+\begin_inset Formula $\alpha\in\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $f\in{\cal C}(\Omega),{\cal H}(\Omega\setminus\{\alpha\})$
+\end_inset
+
+, y 
+\begin_inset Formula $\Delta(a,b,c)\subseteq\Omega$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\int_{[a,b,c,a]}f=0.
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\alpha\notin\Delta(a,b,c)$
+\end_inset
+
+, podemos tomar como abierto 
+\begin_inset Formula $\Omega\setminus\{\alpha\}$
+\end_inset
+
+ y aplicar Cauchy-Goursat.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ es un vértice, por ejemplo, 
+\begin_inset Formula $\alpha=a$
+\end_inset
+
+, sean 
+\begin_inset Formula $c_{\rho}:=(1-\rho)a+\rho b$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{\rho}:=(1-\rho)a+\rho c$
+\end_inset
+
+ para 
+\begin_inset Formula $\rho\in[0,1]$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\int_{[a,b,c,a]}f=\int_{[a,c_{\rho},b_{\rho},a]}f+\int_{[c_{\rho},b,c,c_{\rho}]}+\int_{[c,b_{\rho},c_{\rho},c]}f=\int_{[a,c_{\rho},b_{\rho},a]}f,
+\]
+
+\end_inset
+
+dado que los otros dos sumandos se anulan por el caso anterior.
+ Entonces
+\begin_inset Formula 
+\[
+\left|\int_{[a,b,c,a]}f\right|=\left|\int_{[a,c_{\rho},b_{\rho},a]}f\right|\leq\max_{z\in\Delta(a,b,c)}|f(z)|\rho(|a-b|+|b-c|+|c-a|),
+\]
+
+\end_inset
+
+y haciendo tender 
+\begin_inset Formula $\rho\to0$
+\end_inset
+
+ se obtiene el resultado.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ está en un lado del triángulo, por ejemplo 
+\begin_inset Formula $\alpha\subseteq[a,b]$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\int_{[a,b,c,a]}f=\int_{[a,\alpha,c,a]}f+\int_{[c,\alpha,b,c]}f,
+\]
+
+\end_inset
+
+y cada sumando se anula por el caso anterior.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ está en el interior del triángulo, sea 
+\begin_inset Formula $p$
+\end_inset
+
+ el punto en la intersección de la recta 
+\begin_inset Formula $a\alpha$
+\end_inset
+
+ con 
+\begin_inset Formula $[b,c]$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\int_{[a,b,c,a]}f=\int_{[a,b,p,a]}f+\int_{[a,p,c,a]}f
+\]
+
+\end_inset
+
+y cada sumando se anula por el caso anterior.
+\end_layout
+
+\begin_layout Standard
+De aquí se obtiene el 
+\series bold
+teorema de Cauchy para dominios estrellados 
+\begin_inset Quotes cld
+\end_inset
+
+light
+\begin_inset Quotes crd
+\end_inset
+
+
+\series default
+, que afirma que si 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ es un dominio estrellado en 
+\begin_inset Formula $z_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha\in\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in{\cal C}(\Omega),{\cal H}(\Omega\setminus\{\alpha\})$
+\end_inset
+
+ entonces
+\begin_inset Formula 
+\[
+F(z):=\int_{[z_{0},z]}f
+\]
+
+\end_inset
+
+es una primitiva de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Funciones holomorfas y analíticas
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Fórmula de Cauchy para una circunferencia:
+\series default
+ Sea 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{D}(a,R)\subseteq\Omega$
+\end_inset
+
+, para 
+\begin_inset Formula $z\in D(a,R)$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+f(z)=\frac{1}{2\pi i}\int_{C(a,R)}\frac{f(w)}{w-z}dw.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $\rho>R$
+\end_inset
+
+ con 
+\begin_inset Formula $D(a,\rho)\subseteq\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $z\in D(a,R)$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+g(w):=\begin{cases}
+\frac{f(w)-f(z)}{w-z} & \text{si }w\neq z,\\
+f'(z) & \text{si }w=z.
+\end{cases}
+\]
+
+\end_inset
+
+Como 
+\begin_inset Formula $f$
+\end_inset
+
+ es derivable en 
+\begin_inset Formula $z$
+\end_inset
+
+, 
+\begin_inset Formula $g$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $D(a,\rho)$
+\end_inset
+
+, y es derivable en 
+\begin_inset Formula $D(a,\rho)\setminus\{z\}$
+\end_inset
+
+, luego por el teorema de Cauchy para dominios estrellados 
+\begin_inset Quotes cld
+\end_inset
+
+light
+\begin_inset Quotes crd
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+0=\int_{C(a,R)}g=\int_{C(a,R)}\frac{f(w)-f(z)}{w-z}dw=\int_{C(a,R)}\frac{f(w)}{w-z}dw-f(z)\int_{C(a,R)}\frac{1}{w-z}dw.
+\]
+
+\end_inset
+
+Ahora bien, para 
+\begin_inset Formula $w\in C(a,R)^{*}$
+\end_inset
+
+, como 
+\begin_inset Formula $|z-a|<R$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{|z-a|}{|w-a|}<1$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\frac{1}{w-z}=\frac{1}{w-a-(z-a)}=\frac{1}{w-a}\frac{1}{1-\frac{z-a}{w-a}}=\frac{1}{w-a}\sum_{n}\left(\frac{z-a}{w-a}\right)^{n}=\sum_{n}\frac{|z-a|^{n}}{|w-a|^{n+1}}.
+\]
+
+\end_inset
+
+Pero tomando 
+\begin_inset Formula 
+\[
+\alpha_{n}:=\frac{|z-a|^{n}}{|w-a|^{n+1}}=\frac{1}{R}\left(\frac{|z-a|}{R}\right)^{n},
+\]
+
+\end_inset
+
+como 
+\begin_inset Formula $\frac{|z-a|}{R}<1$
+\end_inset
+
+, la serie 
+\begin_inset Formula $\sum_{n}\alpha_{n}$
+\end_inset
+
+ converge y, por el criterio de Weierstrass, la serie anterior converge
+ uniformemente con 
+\begin_inset Formula $w\in C(a,R)^{*}$
+\end_inset
+
+.
+ Por tanto
+\begin_inset Formula 
+\[
+\int_{C(a,R)}\frac{1}{w-z}dw=\int_{C(a,R)}\sum_{n}\frac{(z-a)^{n}}{(w-a)^{n+1}}dw=\sum_{n}(z-a)^{n}\int_{C(a,R)}\frac{1}{(w-a)^{n+1}}dw,
+\]
+
+\end_inset
+
+pero 
+\begin_inset Formula $w\mapsto\frac{1}{(w-a)^{n+1}}$
+\end_inset
+
+ tiene primitiva 
+\begin_inset Formula $\frac{1}{-n}(w-a)^{-n}$
+\end_inset
+
+ para 
+\begin_inset Formula $n\neq0$
+\end_inset
+
+, luego se anula en 
+\begin_inset Formula $n>0$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\int_{C(a,R)}\frac{1}{w-z}dw=\int_{C(a,R)}\frac{1}{w-a}dw=\int_{-\pi}^{\pi}\frac{1}{a+Re^{it}-a}Rie^{it}dt=\int_{-\pi}^{\pi}idt=2\pi i.
+\]
+
+\end_inset
+
+Sustituyendo,
+\begin_inset Formula 
+\[
+\int_{C(a,R)}\frac{f(w)}{w-z}dw-2\pi if(z)=0,
+\]
+
+\end_inset
+
+y despejando se obtiene el resultado.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Taylor:
+\series default
+ Sean 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+, 
+\begin_inset Formula $\overline{D}(a,R)\subseteq\Omega$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+c_{n}:=\frac{1}{2\pi i}\int_{C(a,R)}\frac{f(w)}{(w-a)^{n+1}}dw
+\]
+
+\end_inset
+
+para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+f(z):=\sum_{n}c_{n}(z-a)^{n}
+\]
+
+\end_inset
+
+para todo 
+\begin_inset Formula $z\in D(a,R)$
+\end_inset
+
+.
+ En particular, 
+\begin_inset Formula $f$
+\end_inset
+
+ es analítica en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $f^{(n)}(a)=n!c_{n}$
+\end_inset
+
+ para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ y los 
+\begin_inset Formula $c_{n}$
+\end_inset
+
+ no dependen del radio escogido.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $z\in D(a,R)$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\frac{f(w)}{w-z}=\frac{f(w)}{w-a-(z-a)}=\frac{f(w)}{w-a}\frac{1}{1-\frac{z-a}{w-a}}=\frac{f(w)}{w-a}\sum_{n}\left(\frac{z-a}{w-a}\right)^{n}=\sum_{n}\frac{f(w)}{(w-a)^{n+1}}(z-a)^{n}.
+\]
+
+\end_inset
+
+Como
+\begin_inset Formula 
+\[
+\frac{|f(w)|}{|w-a|^{n+1}}|z-a|^{n}\leq\alpha_{n}:=\frac{\max_{w\in C(a,R)^{*}}|f(w)|}{R}\left(\frac{|z-a|}{R}\right)^{n}
+\]
+
+\end_inset
+
+y 
+\begin_inset Formula $\sum_{n}\alpha_{n}$
+\end_inset
+
+ es convergente por ser una serie geométrica de razón menor que 1, por el
+ criterio de Weierstrass, la serie converge uniformemente en 
+\begin_inset Formula $C(a,R)^{*}$
+\end_inset
+
+ y, por la fórmula de Cauchy,
+\begin_inset Formula 
+\begin{multline*}
+f(z)=\frac{1}{2\pi i}\int_{C(a,R)}\frac{f(w)}{w-z}dw=\frac{1}{2\pi i}\int_{C(a,R)}\sum_{n}\frac{f(w)}{(w-a)^{n+1}}(z-a)^{n}dw=\\
+=\sum_{n}\left(\frac{1}{2\pi i}\int_{C(a,R)}\frac{f(w)}{(w-a)^{n+1}}dw\right)(z-a)^{n}.
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Morera:
+\series default
+ Sea 
+\begin_inset Formula $f\in{\cal C}(\Omega)$
+\end_inset
+
+, 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+ si y sólo si
+\begin_inset Formula 
+\[
+\int_{[a,b,c,a]}f=0
+\]
+
+\end_inset
+
+para todo 
+\begin_inset Formula $\Delta(a,b,c)\subseteq\Omega$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Teorema de Cauchy-Goursat.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $D(z_{0},R)\subseteq\Omega$
+\end_inset
+
+ un dominio estrellado, como la integral de 
+\begin_inset Formula $f$
+\end_inset
+
+ sobre cualquier triángulo contenido en el disco es 0, por la demostración
+ del teorema de Cauchy para dominios estrellados,
+\begin_inset Formula 
+\[
+F(z)=\int_{[z_{0},z]}f
+\]
+
+\end_inset
+
+es una primitiva de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $D(z_{0},R)$
+\end_inset
+
+, esto es, 
+\begin_inset Formula $\forall z\in D(z_{0},R),F'(z)=f(z)$
+\end_inset
+
+, luego 
+\begin_inset Formula $f$
+\end_inset
+
+ es la derivada de una función holomorfa en 
+\begin_inset Formula $D(z_{0},R)$
+\end_inset
+
+ y en particular 
+\begin_inset Formula $f$
+\end_inset
+
+ es derivable en 
+\begin_inset Formula $z_{0}$
+\end_inset
+
+, pero como 
+\begin_inset Formula $z_{0}\in\Omega$
+\end_inset
+
+ es arbitrario, 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Propiedades de funciones holomorfas
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Desigualdad de Cauchy:
+\series default
+ Sean 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{D}(a,R)\subseteq\Omega$
+\end_inset
+
+, para 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\frac{|f^{(k)}(a)|}{k!}\leq\frac{\max_{w\in C(a,R)^{*}}|f(w)|}{R^{k}}.
+\]
+
+\end_inset
+
+En efecto, tomando módulos sobre la fórmula de la derivada del teorema de
+ Taylor,
+\begin_inset Formula 
+\[
+\frac{|f^{(k)}(a)|}{k!}=\frac{1}{2\pi}\int_{C(a,R)}\frac{f(w)}{(w-a)^{k+1}}dw\leq\frac{1}{2\pi}\max_{w\in C(a,R)^{*}}\left|\frac{f(w)}{(w-a)^{k+1}}\right|2\pi R=\frac{1}{R^{k}}\max_{w\in C(a,R)^{*}}|f(w)|.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Liouville:
+\series default
+ Toda función entera acotada es constante.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $f\in{\cal H}(\mathbb{C})$
+\end_inset
+
+ para la que existe 
+\begin_inset Formula $M>0$
+\end_inset
+
+ con 
+\begin_inset Formula $|f(z)|<M$
+\end_inset
+
+ para todo 
+\begin_inset Formula $z\in\mathbb{C}$
+\end_inset
+
+.
+ Por el teorema de Taylor, para 
+\begin_inset Formula $z\in\mathbb{C}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+f(z)=\sum_{n}\frac{f^{(n)}(0)}{n!}z^{n},
+\]
+
+\end_inset
+
+pero por la desigualdad de Cauchy, para todo 
+\begin_inset Formula $R>0$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\frac{|f^{(n)}(0)|}{n!}\leq\frac{\max_{w\in C(a,R)^{*}}|f(w)|}{R^{n}}\leq\frac{M}{R^{n}},
+\]
+
+\end_inset
+
+y tomando límites cuando 
+\begin_inset Formula $R\to+\infty$
+\end_inset
+
+ tenemos que 
+\begin_inset Formula $f^{(n)}(0)=0$
+\end_inset
+
+ para todo 
+\begin_inset Formula $n\geq1$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f(z)=f(0)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $z$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema fundamental del álgebra:
+\series default
+ 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ es algebraicamente cerrado, esto es, todo polinomio complejo de grado 
+\begin_inset Formula $n$
+\end_inset
+
+ es la forma 
+\begin_inset Formula $p(x)=\alpha\prod_{k=1}^{n}(x-a_{k})$
+\end_inset
+
+ con 
+\begin_inset Formula $\alpha,a_{1},\dots,a_{n}\in\mathbb{C}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Basta ver que todo polinomio complejo no constante tiene alguna raíz, pues
+ el resto se obtiene por inducción.
+ Sea 
+\begin_inset Formula $p$
+\end_inset
+
+ un polinomio de este tipo y supongamos que 
+\begin_inset Formula $\forall z\in\mathbb{C},p(z)\neq0$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $f(z):=\frac{1}{p(z)}$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es entera por serlo 
+\begin_inset Formula $p$
+\end_inset
+
+ y, como 
+\begin_inset Formula $\lim_{z\to+\infty}f(z)=0$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es acotada y, por el teorema de Liouville, constante, y por tanto 
+\begin_inset Formula $p$
+\end_inset
+
+ es constante.
+\begin_inset Formula $\#$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La imagen de una función entera no constante es densa en el plano.
+ 
+\series bold
+Demostración:
+\series default
+ Supongamos que existe 
+\begin_inset Formula $\alpha\in\mathbb{C}\setminus\overline{f(\mathbb{C})}$
+\end_inset
+
+, con lo que existe 
+\begin_inset Formula $\rho>0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\overline{D}(\alpha,\rho)\cap f(\mathbb{C})=\emptyset$
+\end_inset
+
+, esto es, 
+\begin_inset Formula $|f(z)-\alpha|>\rho$
+\end_inset
+
+ para 
+\begin_inset Formula $z\in\mathbb{C}$
+\end_inset
+
+.
+ Sea entonces 
+\begin_inset Formula $g(z):=\frac{1}{f(z)-\alpha}$
+\end_inset
+
+ una función entera, como 
+\begin_inset Formula $|g(z)|=\frac{1}{|f(z)-\alpha|}<\frac{1}{\rho}$
+\end_inset
+
+, 
+\begin_inset Formula $g$
+\end_inset
+
+ es acotada, luego 
+\begin_inset Formula $g$
+\end_inset
+
+ es constante y por tanto 
+\begin_inset Formula $f$
+\end_inset
+
+ también.
+\begin_inset Formula $\#$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de extensión de Riemann:
+\series default
+ Sean 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ un abierto, 
+\begin_inset Formula $\alpha\in\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in{\cal H}(\Omega\setminus\{\alpha\})$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene una extensión holomorfa a 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si y sólo si tiene una extensión continua a 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, si y sólo si está acotada en un entorno reducido de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\lim_{z\to\alpha}(z-\alpha)f(z)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies2\implies3\implies4]$
+\end_inset
+
+ Obvio.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $4\implies1]$
+\end_inset
+
+ Sea 
+\begin_inset Formula 
+\[
+F(z):=\begin{cases}
+(z-\alpha)^{2}f(z) & \text{si }z\neq\alpha,\\
+0 & \text{si }z=\alpha.
+\end{cases}
+\]
+
+\end_inset
+
+
+\begin_inset Formula $F$
+\end_inset
+
+ es holomorfa en 
+\begin_inset Formula $\Omega\setminus\{\alpha\}$
+\end_inset
+
+, pero
+\begin_inset Formula 
+\[
+F'(\alpha)=\lim_{z\to\alpha}\frac{F(z)-F(\alpha)}{z-\alpha}=\lim_{z\to\alpha}(z-\alpha)f(z)=0,
+\]
+
+\end_inset
+
+luego 
+\begin_inset Formula $F\in{\cal H}(\Omega)$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $D(\alpha,\rho)\subseteq\Omega$
+\end_inset
+
+, por el teorema de Taylor, sea 
+\begin_inset Formula $c_{n}:=\frac{F^{(n)}(\alpha)}{n!}$
+\end_inset
+
+, como 
+\begin_inset Formula $c_{0}=c_{1}=0$
+\end_inset
+
+, para 
+\begin_inset Formula $z\in D(\alpha,\rho)$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+F(z)=\sum_{n=2}^{\infty}c_{n}(z-\alpha)^{n}=(z-\alpha)^{2}\sum_{n=2}^{\infty}c_{n}(z-\alpha)^{n-2}=(z-\alpha)^{2}\sum_{n=0}^{\infty}c_{n+2}(z-\alpha)^{n},
+\]
+
+\end_inset
+
+luego si 
+\begin_inset Formula $z\in D(\alpha,\rho)\setminus\{\alpha\}$
+\end_inset
+
+, 
+\begin_inset Formula $f(z)=\sum_{n}c_{n+2}(z-\alpha)^{n}$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula 
+\[
+g(z):=\begin{cases}
+f(z) & \text{si }z\neq\alpha,\\
+c_{2} & \text{si }z=\alpha
+\end{cases}
+\]
+
+\end_inset
+
+es una extensión de 
+\begin_inset Formula $f$
+\end_inset
+
+ expresable como suma de potencias, y por tanto derivable, en 
+\begin_inset Formula $D(\alpha,\rho)$
+\end_inset
+
+, por lo que es derivable en 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ y por tanto una extensión holomorfa de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de convergencia de Weierstrass:
+\series default
+ Sean 
+\begin_inset Formula $\{f_{n}\}_{n}\subseteq{\cal H}(\Omega)$
+\end_inset
+
+, si 
+\begin_inset Formula $(f_{n})_{n}$
+\end_inset
+
+ converge uniformemente en subconjuntos compactos de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $f(z):=\lim_{n}f_{n}(z)$
+\end_inset
+
+ para 
+\begin_inset Formula $z\in\Omega$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es holomorfa en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si y sólo si para cada 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $(f_{n}^{(k)})_{n}$
+\end_inset
+
+ converge uniformemente a 
+\begin_inset Formula $f^{(k)}$
+\end_inset
+
+ en subconjuntos compactos de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Como el límite uniforme de funciones continuas es continuo, 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $\Delta(a,b,c)\subseteq\Omega$
+\end_inset
+
+, como 
+\begin_inset Formula $[a,b,c,a]^{*}$
+\end_inset
+
+ es compacto y la integral respeta la convergencia uniforme,
+\begin_inset Formula 
+\[
+\int_{[a,b,c,a]}f=\lim_{n}\int_{[a,b,c,a]}f_{n}=0
+\]
+
+\end_inset
+
+por el teorema de Cauchy-Goursat, pues las 
+\begin_inset Formula $f_{n}$
+\end_inset
+
+ son holomorfas.
+ Como el triángulo es arbitrario, por el teorema de Morera, 
+\begin_inset Formula $f$
+\end_inset
+
+ es holomorfa en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $K\subseteq\Omega$
+\end_inset
+
+ compacto, 
+\begin_inset Formula $0<\rho<d(K,\partial\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $H:=\{z\in\mathbb{C}:d(z,K)\leq\rho\}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $H$
+\end_inset
+
+ es compacto y 
+\begin_inset Formula $K\subseteq H\subseteq\Omega$
+\end_inset
+
+.
+ Sean 
+\begin_inset Formula $a\in K$
+\end_inset
+
+ y 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+, aplicando la desigualdad de Cauchy a 
+\begin_inset Formula $f_{n}-f$
+\end_inset
+
+ en 
+\begin_inset Formula $\overline{D}(a,\rho)\subseteq H$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+|f_{n}^{(k)}(a)-f^{(k)}(a)|\leq\frac{k!}{\rho^{k}}\max_{w\in C(a,\rho)^{*}}|f_{n}(w)-f(w)|\leq\frac{k!}{\rho^{k}}\max_{w\in H}|f_{n}(w)-f(w)|.
+\]
+
+\end_inset
+
+Por la convergencia uniforme de 
+\begin_inset Formula $(f_{n})_{n}$
+\end_inset
+
+ en 
+\begin_inset Formula $H$
+\end_inset
+
+, dado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, existe 
+\begin_inset Formula $n_{0}\in\mathbb{N}$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $n\geq0$
+\end_inset
+
+ es 
+\begin_inset Formula $\max_{w\in H}|f_{n}(w)-f(w)|\leq\varepsilon$
+\end_inset
+
+, luego 
+\begin_inset Formula $|f_{n}^{(k)}(a)-f^{(k)}(a)|\leq\frac{k!}{\rho^{k}}\varepsilon$
+\end_inset
+
+ para 
+\begin_inset Formula $n\geq n_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in K$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\max_{a\in K}|f_{n}^{(k)}(a)-f^{(k)}(a)|\leq\frac{k!}{\rho^{k}}\varepsilon$
+\end_inset
+
+, de donde 
+\begin_inset Formula $(f_{n}^{(k)})_{n}\to f^{(k)}$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvc/n3.lyx b/fvc/n3.lyx
new file mode 100644
index 0000000..58662d2
--- /dev/null
+++ b/fvc/n3.lyx
@@ -0,0 +1,684 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
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+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ un dominio, 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $Z(f):=\{z\in\Omega:f(z)=0\}$
+\end_inset
+
+, 
+\begin_inset Formula $Z(f)$
+\end_inset
+
+ tiene un punto de acumulación en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\exists a\in\Omega:\forall k\in\mathbb{N},f^{(k)}(a)=0$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $f$
+\end_inset
+
+ es idénticamente nula.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ Sean 
+\begin_inset Formula $a\in Z(f)'\cap\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $D(a,\rho)\subseteq\Omega$
+\end_inset
+
+, existe 
+\begin_inset Formula $\{a_{n}\}_{n}\subseteq D(a,\rho)\setminus\{a\}$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{n}\to a$
+\end_inset
+
+.
+ Por el teorema de Taylor,
+\begin_inset Formula 
+\[
+f(z)=\sum_{n=0}^{\infty}c_{n}(z-a)^{n}
+\]
+
+\end_inset
+
+para 
+\begin_inset Formula $c_{n}:=\frac{f^{(n)}(a)}{n!}$
+\end_inset
+
+, y queremos ver que todos los 
+\begin_inset Formula $c_{n}$
+\end_inset
+
+ son nulos por inducción.
+ Para 
+\begin_inset Formula $n=0$
+\end_inset
+
+, 
+\begin_inset Formula $c_{0}=f(a)=\lim_{n}f(a_{n})=0$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $c_{0}=\dots=c_{k-1}=0$
+\end_inset
+
+, tenemos
+\begin_inset Formula 
+\[
+\frac{f(z)}{(z-a)^{k}}=c_{k}+\sum_{n=k+1}^{\infty}c_{n}(z-a)^{n-k}.
+\]
+
+\end_inset
+
+Sea 
+\begin_inset Formula $g_{k}(z):=\sum_{n=k+1}^{\infty}c_{n}(z-a)^{n-k}$
+\end_inset
+
+ una función holomorfa en 
+\begin_inset Formula $D(a,\rho)$
+\end_inset
+
+ con 
+\begin_inset Formula $g_{k}(a)=0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\frac{f(z)}{(z-a)^{k}}=c_{k}+g_{k}(z)$
+\end_inset
+
+, pero 
+\begin_inset Formula $c_{k}+g_{k}(a_{n})=0$
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+¿por qué?
+\end_layout
+
+\end_inset
+
+ y, tomando límites, 
+\begin_inset Formula $0=c_{k}+\lim_{n}g_{k}(a_{n})=c_{k}+g_{k}(a)=c_{k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies3]$
+\end_inset
+
+ Sea 
+\begin_inset Formula $A:=\{z\in\Omega:\forall k\in\mathbb{N},f^{(k)}(z)=0\}\neq\emptyset$
+\end_inset
+
+, pues 
+\begin_inset Formula $a\in A$
+\end_inset
+
+.
+ Como
+\begin_inset Formula 
+\[
+A=\bigcap_{k=0}^{\infty}\{z\in\Omega:f^{(k)}(z)=0\},
+\]
+
+\end_inset
+
+
+\begin_inset Formula $A$
+\end_inset
+
+ es intersección de cerrados y por tanto cerrado, ahora bien, sean 
+\begin_inset Formula $z\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $D(z,\rho)\subseteq\Omega$
+\end_inset
+
+, por el teorema de Taylor, para 
+\begin_inset Formula $w\in D(z,\rho)$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+f(w)=\sum_{n=0}^{\infty}\frac{f^{(k)}(z)}{n!}(w-z)^{n}=0,
+\]
+
+\end_inset
+
+luego 
+\begin_inset Formula $f^{(k)}(w)=0$
+\end_inset
+
+ y 
+\begin_inset Formula $D(z,\rho)\subseteq A$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $A$
+\end_inset
+
+ es abierto.
+ Por conexión, 
+\begin_inset Formula $A=\Omega$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ Trivial.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+principio de identidad para funciones holomorfas
+\series default
+ afirma que si dos funciones holomorfas en un dominio 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ coinciden en un subconjunto de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ con algún punto de acumulación en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, entonces coinciden en todo 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ En efecto, sean 
+\begin_inset Formula $f,g\in{\cal H}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $h=f-g$
+\end_inset
+
+, si 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ coinciden en un subconjunto de este tipo, entonces 
+\begin_inset Formula $Z(h)'\cap\Omega\neq\emptyset$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $h$
+\end_inset
+
+ es idénticamente nula, luego 
+\begin_inset Formula $f=g$
+\end_inset
+
+.
+ También, si 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ es un dominio y 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+ no es idénticamente nula, entonces todo punto de 
+\begin_inset Formula $Z(f):=\{z\in\Omega:f(z)=0\}$
+\end_inset
+
+ es aislado y 
+\begin_inset Formula $Z(f)$
+\end_inset
+
+ es numerable.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ con 
+\begin_inset Formula $f(a)=0$
+\end_inset
+
+, decimos que 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene en 
+\begin_inset Formula $a$
+\end_inset
+
+ un 
+\series bold
+cero
+\series default
+ de 
+\series bold
+orden
+\series default
+ 
+\begin_inset Formula $\min\{n\in\mathbb{N}:f^{(n)}(a)\neq0\}$
+\end_inset
+
+.
+ Una función 
+\begin_inset Formula $f$
+\end_inset
+
+ holomorfa no idénticamente nula en en un dominio 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ tiene un cero de orden 
+\begin_inset Formula $k\geq1$
+\end_inset
+
+ en 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\exists g\in{\cal H}(\Omega):(g(a)\neq0\land\forall z\in\Omega,f(z)=(z-a)^{k}g(z))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $D(a,\rho)\subseteq\Omega$
+\end_inset
+
+, existe 
+\begin_inset Formula $(c_{n})_{n\geq k}$
+\end_inset
+
+ con 
+\begin_inset Formula $c_{k}\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $f(z)=\sum_{n=k}^{\infty}c_{n}(z-a)^{n}$
+\end_inset
+
+ para cada 
+\begin_inset Formula $z\in D(a,\rho)$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula 
+\[
+g(z):=\begin{cases}
+\frac{f(z)}{(z-a)^{k}} & \text{si }z\neq a,\\
+c_{k} & \text{si }z=a
+\end{cases}
+\]
+
+\end_inset
+
+es holomorfa en 
+\begin_inset Formula $\Omega\setminus\{a\}$
+\end_inset
+
+ y cumple 
+\begin_inset Formula $g(z)=\sum_{n=k}^{\infty}c_{n}(z-a)^{n-k}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $z\in D(a,\rho)$
+\end_inset
+
+, luego es holomorfa en 
+\begin_inset Formula $D(a,\rho)$
+\end_inset
+
+ y, por tanto en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $g\in{\cal H}(\Omega)$
+\end_inset
+
+, 
+\begin_inset Formula $g(a)\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $f(z)=(z-a)^{k}g(z)$
+\end_inset
+
+ para 
+\begin_inset Formula $z\in\Omega$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $D(a,\rho)\subseteq\Omega$
+\end_inset
+
+, por el teorema de Taylor existe 
+\begin_inset Formula $(\alpha_{n})_{n}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $g(z)=\sum_{n}\alpha_{n}(z-a)^{n}$
+\end_inset
+
+ para 
+\begin_inset Formula $z\in D(a,\rho)$
+\end_inset
+
+, con 
+\begin_inset Formula $\alpha_{0}=g(a)\neq0$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $f(z)=\sum_{n}\alpha_{n}(z-a)^{n+k}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f^{(q)}(a)=0$
+\end_inset
+
+ para 
+\begin_inset Formula $q\in\{0,\dots,k-1\}$
+\end_inset
+
+ y 
+\begin_inset Formula $f^{(k)}(a)=k!\alpha_{0}\neq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Regla de L'Hôpital:
+\series default
+ Sean 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ un dominio, 
+\begin_inset Formula $f,g\in{\cal H}(\Omega)$
+\end_inset
+
+ no idénticamente nulas y 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ con 
+\begin_inset Formula $f(a)=g(a)=0$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\lim_{z\to a}\frac{f(z)}{g(z)}=\lim_{z\to a}\frac{f'(z)}{g'(z)}.
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ Por lo anterior, existen 
+\begin_inset Formula $m,n\in\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $F,G\in{\cal H}(\Omega)$
+\end_inset
+
+ con 
+\begin_inset Formula $f(z)=(z-a)^{m}F(z)$
+\end_inset
+
+, 
+\begin_inset Formula $g(z)=(z-a)^{n}G(z)$
+\end_inset
+
+ y 
+\begin_inset Formula $F(a),G(a)\neq0$
+\end_inset
+
+.
+ Como el conjunto de puntos donde 
+\begin_inset Formula $g$
+\end_inset
+
+ se anula está formado por puntos aislados y 
+\begin_inset Formula $g(a)=0$
+\end_inset
+
+, debe haber un disco perforado alrededor de 
+\begin_inset Formula $a$
+\end_inset
+
+ donde 
+\begin_inset Formula $g$
+\end_inset
+
+ no se anula.
+ También hay un disco perforado alrededor de 
+\begin_inset Formula $a$
+\end_inset
+
+ donde 
+\begin_inset Formula $g'$
+\end_inset
+
+ no se anula, por el mismo motivo si 
+\begin_inset Formula $g'(a)=0$
+\end_inset
+
+ o por continuidad si 
+\begin_inset Formula $g'(a)\neq0$
+\end_inset
+
+.
+ Sea entonces 
+\begin_inset Formula $D(a,\rho)\subseteq\Omega$
+\end_inset
+
+ con 
+\begin_inset Formula $g(z)\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $g'(z)\neq0$
+\end_inset
+
+ para 
+\begin_inset Formula $z\in D(a,\rho)\setminus\{a\}$
+\end_inset
+
+, para estos puntos,
+\begin_inset Formula 
+\begin{align*}
+\frac{f(z)}{g(z)} & =\frac{(z-a)^{m}F(z)}{(z-a)^{n}G(z)}, & \frac{f'(z)}{g'(z)} & =(z-a)^{m-n}\frac{mF(z)+(z-a)F'(z)}{nG(z)+(z-a)G'(z)}.
+\end{align*}
+
+\end_inset
+
+Tomando límites cuando 
+\begin_inset Formula $z\to a$
+\end_inset
+
+, si 
+\begin_inset Formula $m=n$
+\end_inset
+
+, ambos límites valen 
+\begin_inset Formula $\frac{F(a)}{G(a)}$
+\end_inset
+
+; si 
+\begin_inset Formula $m>n$
+\end_inset
+
+, ambos son nulos, y si 
+\begin_inset Formula $m<n$
+\end_inset
+
+, ambos son 
+\begin_inset Formula $\infty$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvc/n4.lyx b/fvc/n4.lyx
new file mode 100644
index 0000000..c7ae304
--- /dev/null
+++ b/fvc/n4.lyx
@@ -0,0 +1,2076 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Índice de un punto respecto a una curva
+\end_layout
+
+\begin_layout Standard
+Toda curva 
+\begin_inset Formula $\gamma:[a,b]\to\mathbb{C}^{*}$
+\end_inset
+
+ tiene argumentos continuos, y si 
+\begin_inset Formula $\theta$
+\end_inset
+
+ y 
+\begin_inset Formula $\theta'$
+\end_inset
+
+ son argumentos continuos de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\theta(b)-\theta(a)=\theta'(b)-\theta'(a)$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $\rho:=\min_{t\in[a,b]}|\gamma(t)|>0$
+\end_inset
+
+, 
+\begin_inset Formula $\delta>0$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $|s-t|<\delta$
+\end_inset
+
+ entonces 
+\begin_inset Formula $|\gamma(s)-\gamma(t)|<\rho$
+\end_inset
+
+, 
+\begin_inset Formula $a=t_{0}<\dots<t_{n}=b$
+\end_inset
+
+ una partición de 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ tal que 
+\begin_inset Formula $t_{k}-t_{k-1}<\delta$
+\end_inset
+
+ para cada 
+\begin_inset Formula $k$
+\end_inset
+
+ y 
+\begin_inset Formula $D_{k}:=D(\gamma(t_{k}),\rho)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $0\notin D_{k}$
+\end_inset
+
+ para ningún 
+\begin_inset Formula $k$
+\end_inset
+
+ y 
+\begin_inset Formula $\gamma(t)\in D_{k}$
+\end_inset
+
+ para 
+\begin_inset Formula $t\in[t_{k-1},t_{k}]$
+\end_inset
+
+, luego los discos consecutivos se cortan.
+ Como cada 
+\begin_inset Formula $D_{k}$
+\end_inset
+
+ es un dominio estrellado que no contiene al 0, existe un logaritmo holomorfo,
+ y por tanto un argumento continuo, de la identidad, una función 
+\begin_inset Formula $A_{k}:D_{k}\to\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $A_{k}(z)\in\text{Arg}z$
+\end_inset
+
+ para cada 
+\begin_inset Formula $z\in D_{k}$
+\end_inset
+
+.
+ Sean ahora 
+\begin_inset Formula $\theta_{k}(t):=A_{k}(\gamma(t))\in\text{Arg}(\gamma(t))$
+\end_inset
+
+ y 
+\begin_inset Formula $m_{k}:=\theta_{k}(t_{k})-\theta_{k+1}(t_{k})$
+\end_inset
+
+, y definimos 
+\begin_inset Formula $\theta:[a,b]\to\mathbb{R}$
+\end_inset
+
+ como 
+\begin_inset Formula $\theta(t):=\theta_{k}(t)+\sum_{i=0}^{k-1}m_{k}$
+\end_inset
+
+ para 
+\begin_inset Formula $t\in[t_{k-1},t_{k}]$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\theta$
+\end_inset
+
+ está bien definido, pues 
+\begin_inset Formula $\theta_{k+1}(t_{k})+\sum_{i=0}^{k}m_{k}=\theta_{k+1}(t_{k})+\sum_{i=0}^{k-1}m_{k}+\theta_{k}(t_{k})-\theta_{k+1}(t_{k})=\theta_{k}t_{k}+\sum_{i=0}^{k-1}m_{k}$
+\end_inset
+
+, y es un argumento continuo de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+ Ahora bien, si 
+\begin_inset Formula $\theta$
+\end_inset
+
+ y 
+\begin_inset Formula $\theta'$
+\end_inset
+
+ son argumentos continuos de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+, 
+\begin_inset Formula $\theta-\theta'$
+\end_inset
+
+ es una función continua en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ que toma valores múltiplos de 
+\begin_inset Formula $2\pi$
+\end_inset
+
+ y por tanto debe ser constate, existiendo 
+\begin_inset Formula $k\in\mathbb{Z}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\theta(t)-\theta'(t)=2k\pi$
+\end_inset
+
+ para todo 
+\begin_inset Formula $\theta$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\gamma:[a,b]\to\mathbb{C}$
+\end_inset
+
+ una curva, 
+\begin_inset Formula $z\notin\gamma^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $\theta$
+\end_inset
+
+ un argumento de 
+\begin_inset Formula $\gamma-z$
+\end_inset
+
+, llamamos 
+\series bold
+variación del argumento
+\series default
+ de 
+\begin_inset Formula $z$
+\end_inset
+
+ a lo largo de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ a 
+\begin_inset Formula $\theta(b)-\theta(a)$
+\end_inset
+
+, e 
+\series bold
+índice
+\series default
+ de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ respecto de 
+\begin_inset Formula $z$
+\end_inset
+
+ a
+\begin_inset Formula 
+\[
+\text{Ind}_{\gamma}(z):=\frac{\theta(b)-\theta(a)}{2\pi}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ es una curva cerrada:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{Ind}_{\gamma}:\mathbb{C}\setminus\gamma^{*}\to\mathbb{Z}$
+\end_inset
+
+ es continua, y por tanto constante en cada componente conexa del dominio.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $z_{0}\notin\gamma^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\rho:=\min_{t\in[a,b]}|\gamma(t)-z_{0}|>0$
+\end_inset
+
+ y 
+\begin_inset Formula $z\in D(z_{0},\rho)\subseteq\mathbb{C}\setminus\gamma^{*}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\gamma(t)-z=(\gamma(t)-z_{0})\frac{\gamma(t)-z}{\gamma(t)-z_{0}}$
+\end_inset
+
+, pero 
+\begin_inset Formula 
+\[
+\left|1-\frac{\gamma(t)-z}{\gamma(t)-z_{0}}\right|=\left|\frac{z-z_{0}}{\gamma(t)-z_{0}}\right|<1,
+\]
+
+\end_inset
+
+ luego para 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{\gamma(t)-z}{\gamma(t)-z_{0}}\in D(1,1)$
+\end_inset
+
+, donde el argumento principal es continuo.
+ Sea 
+\begin_inset Formula $\theta_{0}$
+\end_inset
+
+ un argumento de 
+\begin_inset Formula $\gamma-z_{0}$
+\end_inset
+
+, tenemos que 
+\begin_inset Formula $\theta(t):=\theta_{0}(t)+\arg\frac{\gamma(t)-z}{\gamma(t)-z_{0}}$
+\end_inset
+
+ es un argumento continuo de 
+\begin_inset Formula $\gamma(t)-z$
+\end_inset
+
+, pero como 
+\begin_inset Formula $\theta(b)-\theta(a)=\theta_{0}(b)-\theta_{0}(a)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\text{Ind}_{\gamma}(z)=\text{Ind}_{\gamma}(z_{0})$
+\end_inset
+
+ para todo 
+\begin_inset Formula $z\in D(z_{0},\rho)$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $\text{Ind}_{\gamma}$
+\end_inset
+
+ es localmente constante y por tanto continua.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\text{Ind}_{\gamma}$
+\end_inset
+
+ se anula en la única componente no acotada de 
+\begin_inset Formula $\mathbb{C}\setminus\gamma^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+La componente existe y es única porque 
+\begin_inset Formula $\gamma^{*}$
+\end_inset
+
+, al ser la imagen de un compacto por una función continua, es un compacto
+ y existe 
+\begin_inset Formula $R$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\gamma^{*}\subseteq D(0,R)$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\mathbb{C}\setminus D(0,R)\subseteq\mathbb{C}\setminus\gamma^{*}$
+\end_inset
+
+, y al ser conexo, está en una componente conexa del conjunto.
+ Sea ahora 
+\begin_inset Formula $z_{0}$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{Re}z_{0}<-R$
+\end_inset
+
+, es claro que 
+\begin_inset Formula $z_{0}\notin\gamma^{*}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\text{Re}(\gamma(t)-z_{0})>0$
+\end_inset
+
+ y, como 
+\begin_inset Formula $(\gamma-z_{0})^{*}$
+\end_inset
+
+ está en el semiplano de la derecha, el argumento principal es continuo.
+ Como 
+\begin_inset Formula $z_{0}$
+\end_inset
+
+ está en la componente conexa no acotada y el índice es constante en cada
+ componente, para 
+\begin_inset Formula $z$
+\end_inset
+
+ en la componente no acotada, 
+\begin_inset Formula $\text{Ind}_{\gamma}z=\text{Ind}_{\gamma}z_{0}=\frac{\arg(\gamma(b)-z_{0})-\arg(\gamma(a)-z_{0})}{2\pi}=0$
+\end_inset
+
+.
+ 
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ un camino cerrado y 
+\begin_inset Formula $z\notin\gamma^{*}$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\text{Ind}_{\gamma}(z)=\frac{1}{2\pi i}\int_{\gamma}\frac{1}{w-z}dw.
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $\theta$
+\end_inset
+
+ un argumento continuo de 
+\begin_inset Formula $\gamma-z$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\varphi(t):=\log|\gamma(t)-z|+i\theta(t)$
+\end_inset
+
+ es un logaritmo continuo de 
+\begin_inset Formula $\gamma(t)-z$
+\end_inset
+
+ .
+ Sea 
+\begin_inset Formula $a=t_{0}<\dots<t_{n}=b$
+\end_inset
+
+ una partición del dominio 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\gamma|_{[t_{k-1},t_{k}]}$
+\end_inset
+
+ es derivable.
+ Entonces 
+\begin_inset Formula $\varphi_{k}:=\varphi|_{[t_{k-1},t_{k}]}$
+\end_inset
+
+ también lo es y 
+\begin_inset Formula $\varphi'_{k}(t)=\frac{\gamma'(t)}{\gamma(t)-z}$
+\end_inset
+
+ para 
+\begin_inset Formula $t\in(t_{k-1},t_{k})$
+\end_inset
+
+.
+ Integrando,
+\begin_inset Formula 
+\begin{multline*}
+\int_{\gamma}\frac{1}{w-z}dw=\int_{a}^{b}\frac{\gamma'(t)}{\gamma(t)-z}dt=\sum_{k=1}^{n}\int_{t_{k-1}}^{t_{k}}\frac{\gamma'(t)}{\gamma(t)-z}dt=\sum_{k=1}^{n}(\varphi_{k}(t_{k})-\varphi_{k}(t_{k-1}))=\\
+=\sum_{k=1}^{n}(\varphi(t_{k})-\varphi(t_{k-1}))=\varphi(b)-\varphi(a)=\log|\gamma(b)-z|+i\theta(b)-\log|\gamma(a)-z|-i\theta(a)=\\
+=i(\theta(b)-\theta(a))=2\pi i\text{Ind}_{\gamma}(z).
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Cadenas
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+cadena
+\series default
+ es una expresión de la forma 
+\begin_inset Formula $\Gamma:=m_{1}\gamma_{1}+\dots+m_{q}\gamma_{q}$
+\end_inset
+
+ donde los 
+\begin_inset Formula $m_{i}$
+\end_inset
+
+ son enteros y los 
+\begin_inset Formula $\gamma_{i}$
+\end_inset
+
+ son caminos.
+ Llamamos 
+\series bold
+soporte
+\series default
+ de 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ a 
+\begin_inset Formula $\Gamma^{*}:=\bigcup_{k}\gamma_{k}^{*}$
+\end_inset
+
+ y 
+\series bold
+longitud
+\series default
+ de 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ a 
+\begin_inset Formula $\ell(\Gamma):=\sum_{k}|m_{k}|\ell(\gamma_{k})$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\Sigma:=n_{1}\sigma_{1}+\dots+n_{p}\sigma_{p}$
+\end_inset
+
+ es otra cadena, llamamos 
+\begin_inset Formula $\Gamma+\Sigma:=m_{1}\gamma_{1}+\dots+m_{q}\gamma_{q}+k_{1}\sigma_{1}+\dots+k_{p}\sigma_{p}$
+\end_inset
+
+.
+ Dada 
+\begin_inset Formula $f:\Gamma^{*}\to\mathbb{C}$
+\end_inset
+
+, llamamos
+\begin_inset Formula 
+\[
+\int_{\Gamma}f:=\sum_{k}m_{k}\int_{\gamma_{k}}f,
+\]
+
+\end_inset
+
+y claramente se cumplen la aditividad de la integral y la acotación básica
+\begin_inset Formula 
+\[
+\left|\int_{\Gamma}f\right|\leq\ell(\Gamma)\max_{z\in\Gamma^{*}}|f(z)|.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+ciclo
+\series default
+ es una cadena formada por caminos cerrados, y llamamos 
+\series bold
+índice
+\series default
+ de 
+\begin_inset Formula $z\notin\Gamma^{*}$
+\end_inset
+
+ respecto al ciclo 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{Ind}_{\Gamma}(z):=\sum_{k}m_{k}\text{Ind}_{\gamma_{k}}(z)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\text{Ind}_{\Gamma}:\mathbb{C}\setminus\Gamma^{*}\to\mathbb{Z}$
+\end_inset
+
+ es continua y constante en cada componente conexa del dominio, y se anula
+ en la componente conexa no acotada.
+\end_layout
+
+\begin_layout Standard
+Dos cadenas 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ y 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+ son 
+\series bold
+equivalentes
+\series default
+ si para toda 
+\begin_inset Formula $f$
+\end_inset
+
+ continua en 
+\begin_inset Formula $\Gamma^{*}\cup\Sigma^{*}$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+\int_{\Gamma}f=\int_{\Sigma}f.
+\]
+
+\end_inset
+
+Dado un abierto 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, un ciclo 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ es 
+\series bold
+nulhomólogo
+\series default
+ respecto de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si 
+\begin_inset Formula $\forall z\in\mathbb{C}\setminus\Omega,\text{Ind}_{\Gamma}(z)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Forma general del teorema de Cauchy
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Fórmula integral de Cauchy:
+\series default
+ Sean 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ un ciclo en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ nulhomólogo respecto de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in{\cal H}(\Omega)$
+\end_inset
+
+, para 
+\begin_inset Formula $z\in\Omega\setminus\Gamma^{*}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+f(z)\text{Ind}_{\Gamma}(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{f(w)}{w-z}dw.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Primero vemos que si 
+\begin_inset Formula $F:\Omega\times\Gamma^{*}\to\mathbb{C}$
+\end_inset
+
+ es continua, entonces 
+\begin_inset Formula $h:\Omega\to\mathbb{C}$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+h(z):=\int_{\Gamma}F(z,w)dw
+\]
+
+\end_inset
+
+es continua en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ En efecto, sean 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $(z_{n})_{n}$
+\end_inset
+
+ una sucesión de puntos de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ convergente a 
+\begin_inset Formula $a$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+|h(z_{n})-h(a)|=\left|\int_{\Gamma}(F(z_{n},w)-F(a,w))dw\right|\leq\ell(\Gamma)\max_{w\in\Gamma^{*}}|F(z_{n},w)-F(a,w)|.
+\]
+
+\end_inset
+
+Como 
+\begin_inset Formula $K:=\{\{z_{n}\}_{n}\cup\{a\}\}\times\Gamma^{*}$
+\end_inset
+
+ es compacto por ser producto de compactos, 
+\begin_inset Formula $F$
+\end_inset
+
+ es uniformemente continua en 
+\begin_inset Formula $K$
+\end_inset
+
+.
+ Dado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, existe 
+\begin_inset Formula $\delta>0$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $|z-z'|,|w-w'|<\delta$
+\end_inset
+
+ entonces 
+\begin_inset Formula $|F(z,w)-F(z',w')|<\varepsilon$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $n_{0}$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $n\geq n_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $|z_{n}-z_{0}|<\delta$
+\end_inset
+
+, entonces 
+\begin_inset Formula $|F(z_{n},w)-F(a,w)|<\varepsilon$
+\end_inset
+
+, luego 
+\begin_inset Formula $|h(z_{n})-h(a)|\leq\ell(\Gamma)\varepsilon$
+\end_inset
+
+ para 
+\begin_inset Formula $n\geq n_{0}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $h(z_{n})\to h(a)$
+\end_inset
+
+ y, como 
+\begin_inset Formula $a$
+\end_inset
+
+ es arbitrario, 
+\begin_inset Formula $h$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Si además, para 
+\begin_inset Formula $w\in\Gamma^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $F_{w}:\Omega\to\mathbb{C}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $F_{w}(z):=F(z,w)$
+\end_inset
+
+ es holomorfa en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, entonces 
+\begin_inset Formula $h\in{\cal H}(\Omega)$
+\end_inset
+
+.
+ Primero vemos que, dados 
+\begin_inset Formula $\gamma:[a,b]\to\mathbb{C}$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma:[c,d]\to\mathbb{C}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{multline*}
+\int_{\sigma}\int_{\gamma}F(z,w)dw\,dz=\int_{c}^{d}\int_{a}^{b}F(\sigma(s),\gamma(t))\gamma'(t)dt\,\sigma'(s)ds=\\
+=\int_{a}^{b}\int_{c}^{d}F(\sigma(s),\gamma(t))\sigma'(s)ds\,\gamma'(t)dt=\int_{\gamma}\int_{\sigma}F(z,w)dz\,dw,
+\end{multline*}
+
+\end_inset
+
+y por linealidad esto también sirve cuando en vez de curvas tenemos cadenas.
+ Entonces, para 
+\begin_inset Formula $\Delta(a,b,c)\subseteq\Omega$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\int_{[a,b,c,a]}h=\int_{[a,b,c,a]}\int_{\Gamma}F(z,w)dw\,dz=\int_{\Gamma}\int_{[a,b,c,a]}F(z,w)dz\,dw=\int_{\Gamma}0\,dw=0,
+\]
+
+\end_inset
+
+pues 
+\begin_inset Formula $F_{w}(z)=F(z,w)$
+\end_inset
+
+ es holomorfa en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ Como el triángulo era arbitrario, el teorema de Morera nos dice que 
+\begin_inset Formula $h\in{\cal H}(\Omega)$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Sea ahora 
+\begin_inset Formula 
+\[
+F(z,w):=\begin{cases}
+\frac{f(w)-f(z)}{w-z} & \text{si }z\neq w,\\
+f'(z) & \text{si }z=w.
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $F$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $\{(z,w)\in\Omega\times\Omega:z\neq w\}$
+\end_inset
+
+.
+ Para ver que también lo es en los puntos de la forma 
+\begin_inset Formula $(a,a)$
+\end_inset
+
+ con 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+, dado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, existe 
+\begin_inset Formula $\delta>0$
+\end_inset
+
+ con 
+\begin_inset Formula $D(a,\delta)\subseteq\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $|f'(z)-f'(a)|<\varepsilon$
+\end_inset
+
+ para 
+\begin_inset Formula $z\in D(a,\delta)$
+\end_inset
+
+, y queremos ver que, si 
+\begin_inset Formula $|z-a|,|w-a|<\delta$
+\end_inset
+
+, entonces 
+\begin_inset Formula $|F(z,w)-F(a,a)|=|F(z,w)-f'(a)|<\varepsilon$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $z=w$
+\end_inset
+
+, esto equivale a que 
+\begin_inset Formula $|f'(z)-f'(a)|<\varepsilon$
+\end_inset
+
+, que se cumple por hipótesis.
+ Para 
+\begin_inset Formula $z\neq w$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+f(z)-f(w)=\int_{[w,z]}f'(u)du=\int_{0}^{1}f'((1-t)w+tz)(z-w)dt,
+\]
+
+\end_inset
+
+luego 
+\begin_inset Formula 
+\[
+F(z,w)=\int_{0}^{1}f'((1-t)w+tz)dt.
+\]
+
+\end_inset
+
+Entonces
+\begin_inset Formula 
+\[
+|F(z,w)-f'(a)|=\left|\int_{0}^{1}(f'((1-t)w+tz)-f'(a))dt\right|\leq\int_{0}^{1}|f'((1-t)w+tz)-f'(a)|dt<\varepsilon,
+\]
+
+\end_inset
+
+pues 
+\begin_inset Formula $(1-t)w+tz\in[w,z]^{*}\subseteq D(a,\delta)$
+\end_inset
+
+ y podemos usar esta acotación.
+\end_layout
+
+\begin_layout Standard
+Ahora bien, fijado 
+\begin_inset Formula $w\in\Omega$
+\end_inset
+
+, sea 
+\begin_inset Formula $F_{w}(z):=F(w,z)$
+\end_inset
+
+, es claro que 
+\begin_inset Formula $F_{w}\in{\cal H}(\Omega\setminus\{w\})$
+\end_inset
+
+, y como 
+\begin_inset Formula $F_{w}\in{\cal C}(\Omega)$
+\end_inset
+
+, por el teorema de extensión de Riemann, 
+\begin_inset Formula $F_{w}\in{\cal H}(\Omega)$
+\end_inset
+
+, de donde 
+\begin_inset Formula $h\in{\cal H}(\Omega)$
+\end_inset
+
+ por el resultado de antes.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\Omega_{0}:=\{z\in\mathbb{C}\setminus\Gamma^{*}:\text{Ind}_{\Gamma}(z)=0\}$
+\end_inset
+
+, que es abierto por ser unión de componentes conexas de 
+\begin_inset Formula $\mathbb{C}\setminus\Gamma^{*}$
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+¿por qué las componentes son abiertas?
+\end_layout
+
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ es nulhomólogo respecto a 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{C}\setminus\Omega\subseteq\Omega_{0}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\Omega\cup\Omega_{0}=\mathbb{C}$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $F_{0}:\Omega_{0}\times\Gamma^{*}\to\mathbb{C}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $F_{0}(z,w):=\frac{f(w)}{w-z}$
+\end_inset
+
+.
+ Esta está bien definida por ser 
+\begin_inset Formula $\Omega_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Gamma^{*}$
+\end_inset
+
+ disjuntos y por tanto 
+\begin_inset Formula $z\neq w$
+\end_inset
+
+, y es continua.
+ Fijado 
+\begin_inset Formula $w\in\Omega^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $F_{0w}\in{\cal H}(\Omega_{0})$
+\end_inset
+
+ por ser una función racional, con lo que por el resultado del principio,
+\begin_inset Formula 
+\[
+h_{0}(z):=\int_{\Gamma}F_{0}(z,w)dw=\int_{\Gamma}\frac{f(w)}{w-z}dw
+\]
+
+\end_inset
+
+es holomorfa en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ Sea ahora
+\begin_inset Formula 
+\[
+\varphi(z):=\begin{cases}
+h(z) & \text{si }z\in\Omega,\\
+h_{0}(z) & \text{si }z\in\Omega_{0},
+\end{cases}
+\]
+
+\end_inset
+
+para ver que 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ está bien definida debemos ver que para 
+\begin_inset Formula $z\in\Omega\cap\Omega_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $h(z)=h_{0}(z)$
+\end_inset
+
+, pero como 
+\begin_inset Formula $z\in\Omega_{0}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $z\notin\Gamma^{*}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{multline*}
+h(z)=\int_{\Gamma}\frac{f(w)-f(z)}{w-z}dw=\int_{\Gamma}\frac{f(w)}{w-z}dw-f(z)\int_{\Gamma}\frac{1}{w-z}dw=\\
+=\int_{\Gamma}\frac{f(w)}{w-z}dw-f(z)\text{Ind}_{\Gamma}(z)2\pi i\overset{\text{Ind}_{\gamma}(z)=0}{=}h_{0}(z).
+\end{multline*}
+
+\end_inset
+
+Como 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es holomorfa en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $\Omega_{0}$
+\end_inset
+
+, es entera.
+ Sea 
+\begin_inset Formula $R>0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\Gamma^{*}\subseteq D(0,R)$
+\end_inset
+
+ y 
+\begin_inset Formula $|z|>R$
+\end_inset
+
+, entonces 
+\begin_inset Formula $z$
+\end_inset
+
+ está en la componente no acotada de 
+\begin_inset Formula $\mathbb{C}\setminus\Gamma^{*}$
+\end_inset
+
+ y por tanto en 
+\begin_inset Formula $\Omega_{0}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\varphi(z)=h_{0}(z)$
+\end_inset
+
+, y como 
+\begin_inset Formula $\forall w\in\Gamma^{*},|w-z|\geq|z|-R$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+|\varphi(z)|=\left|\int_{\Gamma}\frac{f(w)}{w-z}dw\right|\leq\ell(\Gamma)\max_{w\in\Gamma^{*}}\left|\frac{f(w)}{w-z}\right|\leq\ell(\Gamma)\frac{\max_{w\in\Gamma^{*}}|f(w)|}{|z|-R}.
+\]
+
+\end_inset
+
+Tomando límites cuando 
+\begin_inset Formula $z\to\infty$
+\end_inset
+
+ queda 
+\begin_inset Formula $\lim_{z\to\infty}\varphi(z)=0$
+\end_inset
+
+, luego 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ está acotada y, por el teorema de Liouville, es constante, y como el límite
+ vale 0, es idénticamente nula.
+ Entonces, para 
+\begin_inset Formula $z\in\Omega\setminus\Gamma^{*}$
+\end_inset
+
+, 
+\begin_inset Formula 
+\begin{multline*}
+0=\varphi(z)=h(z)=\int_{\Gamma}\frac{f(w)-f(z)}{w-z}dw=\int_{\Gamma}\frac{f(w)}{w-z}dw-f(z)\int_{\Gamma}\frac{1}{w-z}dw=\\
+=\int_{\Gamma}\frac{f(w)}{w-z}dw-f(z)\text{Ind}_{\Gamma}(z)2\pi i,
+\end{multline*}
+
+\end_inset
+
+y despejando se obtiene la fórmula.
+\end_layout
+
+\begin_layout Standard
+En estas condiciones, la 
+\series bold
+forma general del teorema de Cauchy
+\series default
+ afirma que
+\begin_inset Formula 
+\[
+\int_{\Gamma}f=0.
+\]
+
+\end_inset
+
+ En efecto, para 
+\begin_inset Formula $a\in\Omega\setminus\Gamma^{*}$
+\end_inset
+
+, aplicando la fórmula integral de Cauchy a 
+\begin_inset Formula $g(z):=(z-a)f(z)$
+\end_inset
+
+, como 
+\begin_inset Formula $g(a)=0$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+0=g(a)\text{Ind}_{\Gamma}(a)=\frac{1}{2\pi i}\int_{\Gamma}\frac{(w-a)f(w)}{w-a}dw=\frac{1}{2\pi i}\int_{\Gamma}f.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dos ciclos 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ y 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+ en un abierto 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ son 
+\series bold
+homológicamente equivalentes
+\series default
+ respecto de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si 
+\begin_inset Formula $\forall z\in\mathbb{C}\setminus\Omega,\text{Ind}_{\Gamma}(z)=\text{Ind}_{\Sigma}(z)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ es 
+\series bold
+homológicamente conexo
+\series default
+ si todo ciclo en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ es nulhomólogo respecto de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, si y sólo si es un abierto cuyo complemento no tiene componentes conexas
+ acotadas.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ un abierto de este tipo, 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ un camino cerrado en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $R>0$
+\end_inset
+
+ con 
+\begin_inset Formula $\gamma^{*}\subseteq D(0,R)$
+\end_inset
+
+, 
+\begin_inset Formula $z\notin\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $C$
+\end_inset
+
+ la componente conexa de 
+\begin_inset Formula $\mathbb{C}\setminus\Omega$
+\end_inset
+
+ que contiene a 
+\begin_inset Formula $z$
+\end_inset
+
+ y 
+\begin_inset Formula $U$
+\end_inset
+
+ la componente conexa no acotada de 
+\begin_inset Formula $\mathbb{C}\setminus\gamma^{*}$
+\end_inset
+
+.
+ Claramente 
+\begin_inset Formula $\mathbb{C}\setminus D(0,R)\subseteq U$
+\end_inset
+
+, y como por hipótesis, 
+\begin_inset Formula $C$
+\end_inset
+
+ no está acotada, debe ser 
+\begin_inset Formula $C\cap U\neq\emptyset$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $C\subseteq\mathbb{C}\setminus\Omega\subseteq\mathbb{C}\setminus\gamma^{*}$
+\end_inset
+
+, debe ser 
+\begin_inset Formula $C\subseteq U$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $z\in U$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\text{Ind}_{\gamma}(z)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Singularidades aisladas
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ un abierto, 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in{\cal H}(\Omega\setminus\{a\})$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+regular
+\series default
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ o 
+\begin_inset Formula $a$
+\end_inset
+
+ es un 
+\series bold
+punto regular
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ si podemos definir 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ de forma que sea derivable en 
+\begin_inset Formula $a$
+\end_inset
+
+, si y solo si 
+\begin_inset Formula $\lim_{z\to a}f(z)\in\mathbb{C}$
+\end_inset
+
+ (suponiendo que dicho límite exista) por el teorema de extensión de Riemann,
+ y de lo contrario decimos que 
+\begin_inset Formula $a$
+\end_inset
+
+ es un 
+\series bold
+punto singular
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ o 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene una 
+\series bold
+singularidad aislada
+\series default
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un punto singular 
+\begin_inset Formula $a$
+\end_inset
+
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ es un 
+\series bold
+polo
+\series default
+ de orden 
+\begin_inset Formula $k$
+\end_inset
+
+ si 
+\begin_inset Formula $k$
+\end_inset
+
+ es el mínimo natural tal que 
+\begin_inset Formula $z\mapsto(z-a)^{k}f(z)$
+\end_inset
+
+ es regular, y si no existe tal 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $a$
+\end_inset
+
+ es una 
+\series bold
+singularidad esencial
+\series default
+.
+ La función 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene un polo en 
+\begin_inset Formula $a$
+\end_inset
+
+ de orden 
+\begin_inset Formula $k$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\lim_{z\to a}(z-a)^{k}f(z)\in\mathbb{C}^{*}$
+\end_inset
+
+ (suponiendo que el límite exista), si y sólo si 
+\begin_inset Formula $\exists\varphi\in{\cal H}(\Omega):(\varphi(a)\neq0\land\forall z\in\Omega\setminus\{a\},f(z)=\frac{\varphi(z)}{(z-a)^{k}})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados un abierto 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in{\cal H}(\Omega\setminus\{a\})$
+\end_inset
+
+, llamamos 
+\series bold
+residuo
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ a
+\begin_inset Formula 
+\[
+\text{Res}(f,a):=\frac{1}{2\pi i}\int_{C(a,\rho)}f,
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $\rho$
+\end_inset
+
+ es cualquier radio tal que 
+\begin_inset Formula $D(a,\rho)\setminus\{a\}\subseteq\Omega$
+\end_inset
+
+.
+ El valor no depende del radio, pues 
+\begin_inset Formula 
+\begin{multline*}
+\int_{C(a,R)}f-\int_{C(a,\rho)}f=\\
+=\int_{C(a,R)|_{[0,\pi]}\dot{+}[-R,-\rho]\dot{-}C(a,\rho)|_{[0,\pi]}\dot{+}[\rho,R]}f+\int_{C(a,R)|_{[-\pi,0]}\dot{+}[R,\rho]\dot{-}C(a,\rho)|_{[-\pi,0]}\dot{+}[-\rho,-R]}f=0+0=0,
+\end{multline*}
+
+\end_inset
+
+incluyendo cada curva en un abierto nulhomólogo que contenga al semianillo.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a$
+\end_inset
+
+ es regular, 
+\begin_inset Formula $\text{Res}(f,a)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Basta extender 
+\begin_inset Formula $f$
+\end_inset
+
+ a 
+\begin_inset Formula $a$
+\end_inset
+
+ de forma holomorfa y aplicar el teorema de Cauchy.
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a$
+\end_inset
+
+ es un polo de orden 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+\text{Res}(f,a)=\frac{1}{(k-1)!}\lim_{z\to a}\left((z-a)^{k}f(z)\right)^{(k-1)}.
+\]
+
+\end_inset
+
+Existe 
+\begin_inset Formula $g\in{\cal H}(\Omega)$
+\end_inset
+
+ con 
+\begin_inset Formula $g(z)=(z-a)^{k}f(z)$
+\end_inset
+
+ para 
+\begin_inset Formula $z\in\Omega\setminus\{a\}$
+\end_inset
+
+.
+ Cerca de 
+\begin_inset Formula $a$
+\end_inset
+
+, tendrá una serie de Taylor 
+\begin_inset Formula $\sum_{n}c_{n}(z-a)^{n}$
+\end_inset
+
+, pero por el teorema de Taylor, 
+\begin_inset Formula 
+\[
+\frac{g^{(n)}(a)}{n!}=c_{n}=\frac{1}{2\pi i}\int_{C(a,\rho)}\frac{g(z)}{(z-a)^{n+1}}dz,
+\]
+
+\end_inset
+
+y en particular
+\begin_inset Formula 
+\[
+\text{Res}(f,a)=\frac{1}{2\pi i}\int_{C(a,\rho)}f=\frac{1}{2\pi i}\int_{C(a,\rho)}\frac{g(z)}{(z-a)^{k}}dz=c_{k-1}=\frac{g^{(k-1)}(a)}{(k-1)!}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $f=\frac{g}{h}$
+\end_inset
+
+ con 
+\begin_inset Formula $g,h\in{\cal H}(\Omega)$
+\end_inset
+
+, las únicas singularidades de 
+\begin_inset Formula $f$
+\end_inset
+
+ son los ceros de 
+\begin_inset Formula $h$
+\end_inset
+
+, y si 
+\begin_inset Formula $h$
+\end_inset
+
+ tiene un cero de orden 
+\begin_inset Formula $k$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $g(a)\neq0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene un polo de orden 
+\begin_inset Formula $k$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Si el polo es simple, 
+\begin_inset Formula 
+\[
+\text{Res}(f,a)=\frac{g(a)}{h'(a)}.
+\]
+
+\end_inset
+
+
+\begin_inset Formula $\text{Res}(f,a)=\lim_{z\to a}(z-a)f(z)=\lim_{z\to a}(z-a)\frac{g(z)}{h(z)}=g(a)\lim_{z\to a}\frac{z-a}{h(z)}=\frac{g(a)}{h'(a)}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Teorema de los residuos
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\Omega\subseteq\mathbb{C}$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $S\subseteq\Omega$
+\end_inset
+
+ con 
+\begin_inset Formula $S'\cap\Omega=\emptyset$
+\end_inset
+
+, 
+\begin_inset Formula $f\in{\cal H}(\Omega\setminus S)$
+\end_inset
+
+ y 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ un ciclo en 
+\begin_inset Formula $\Omega\setminus S$
+\end_inset
+
+ nulhomólogo respecto de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\{a\in S:\text{Ind}_{\Gamma}(a)\neq0\}$
+\end_inset
+
+ es finito y
+\begin_inset Formula 
+\[
+\int_{\Gamma}f=2\pi i\sum_{a\in S}\text{Res}(f,a)\text{Ind}_{\Gamma}(a).
+\]
+
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $\Omega_{0}=\{z\in\mathbb{C}\setminus\Gamma^{*}:\text{Ind}_{\Gamma}(z)=0\}$
+\end_inset
+
+, que es abierto por ser unión de componentes conexas de 
+\begin_inset Formula $\mathbb{C}\setminus\Gamma^{*}$
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+¿por qué?
+\end_layout
+
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ es nulhomólogo respecto de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{C}\setminus\Omega\subseteq\Omega_{0}$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $K:=\mathbb{C}\setminus\Omega_{0}=\Gamma^{*}\cup\{z\in\mathbb{C}\setminus\Gamma^{*}:\text{Ind}_{\Gamma}(z)\neq0\}$
+\end_inset
+
+, que es cerrado por ser complementario de un abierto y acotado porque no
+ corta a la componente no acotada de 
+\begin_inset Formula $\mathbb{C}\setminus\Gamma^{*}$
+\end_inset
+
+, luego es compacto.
+ Si 
+\begin_inset Formula $S\cap K=\{a\in S:\text{Ind}_{\Gamma}(z)\neq0\}$
+\end_inset
+
+ no fuera finito, tendría un punto de acumulación que, por compacidad, debería
+ quedarse en 
+\begin_inset Formula $K\subseteq\Omega$
+\end_inset
+
+, luego no sería 
+\begin_inset Formula $S'\cap\Omega=\emptyset\#$
+\end_inset
+
+.
+ Así, la suma en el enunciado del teorema es finita.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $S\cap K=:\{a_{1},\dots,a_{q}\}$
+\end_inset
+
+, 
+\begin_inset Formula $\rho>0$
+\end_inset
+
+ tal que para cada 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $\overline{D}(a_{k},\rho)\subseteq\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{D}(a_{k},\rho)\cap S=\{a_{k}\}$
+\end_inset
+
+, 
+\begin_inset Formula $m_{k}:=\text{Ind}_{\Gamma}(a_{k})$
+\end_inset
+
+, 
+\begin_inset Formula $\gamma_{k}:=C(a_{k},\rho)$
+\end_inset
+
+ y 
+\begin_inset Formula $\Sigma:=\sum_{k=1}^{q}m_{k}\gamma_{k}$
+\end_inset
+
+.
+ Veamos que 
+\begin_inset Formula $\Gamma-\Sigma$
+\end_inset
+
+ es nulhomólogo respecto de 
+\begin_inset Formula $\Omega\setminus S$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $z\notin\Omega\setminus S$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $z\notin\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ind}_{\Gamma}(z)=0$
+\end_inset
+
+, y como 
+\begin_inset Formula $z\notin\overline{D}(a_{k},\rho)$
+\end_inset
+
+ para ningún 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ind}_{\Sigma}(z)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $z\in S$
+\end_inset
+
+, 
+\begin_inset Formula $z\neq a_{1},\dots,a_{q}$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ind}_{\Gamma}(z)=0$
+\end_inset
+
+ y como, por definición, 
+\begin_inset Formula $z\notin\overline{D}(a_{k},\rho)$
+\end_inset
+
+ para ningún 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ind}_{\Sigma}(z)=0$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $z=a_{j}$
+\end_inset
+
+ para algún 
+\begin_inset Formula $j$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ind}_{\Gamma}(a_{j})=m_{j}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\text{Ind}_{\gamma_{j}}(a_{j})=1$
+\end_inset
+
+, y para 
+\begin_inset Formula $k\neq j$
+\end_inset
+
+, 
+\begin_inset Formula $a_{j}\notin\overline{D}(a_{k},\rho)$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\text{Ind}_{\gamma_{k}}(a_{j})=0$
+\end_inset
+
+, luego 
+\begin_inset Formula $\text{Ind}_{\Sigma}(a_{j})=m_{j}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Aplicando ahora el teorema de Cauchy,
+\begin_inset Formula 
+\[
+\int_{\Gamma-\Sigma}f=0,
+\]
+
+\end_inset
+
+luego
+\begin_inset Formula 
+\[
+\int_{\Gamma}f=\int_{\Sigma}f=\sum_{k=1}^{q}m_{k}\int_{\gamma_{k}}f=\sum_{k=1}^{q}\text{Ind}_{\Gamma}(a_{j})\int_{C(a_{j},\rho)}f=2\pi i\sum_{k=1}^{q}\text{Ind}_{\Gamma}(a_{j})\text{Res}(f,a_{j}).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document