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+#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 swiss
+\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
+En este capítulo, 
+\begin_inset Formula $K$
+\end_inset
+
+ representa indistintamente a 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ o 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+.
+ Una 
+\series bold
+serie de potencias
+\series default
+ en torno a 
+\begin_inset Formula $z_{0}\in K$
+\end_inset
+
+ es una expresión de la forma
+\begin_inset Formula 
+\[
+\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $(a_{n})_{n=0}^{\infty}$
+\end_inset
+
+ es una sucesión de elementos de 
+\begin_inset Formula $K$
+\end_inset
+
+ y 
+\begin_inset Formula $z\in K$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+radio de convergencia
+\series default
+ de la serie al valor
+\begin_inset Formula 
+\[
+R:=\frac{1}{\limsup_{n}\sqrt[n]{|a_{n}|}}
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $\limsup_{n}a_{n}$
+\end_inset
+
+ es el supremo de las subsucesiones convergentes de 
+\begin_inset Formula $(a_{n})$
+\end_inset
+
+.
+ Se entiende que si 
+\begin_inset Formula $\limsup_{n}\sqrt[n]{|a_{n}|}=0$
+\end_inset
+
+ se toma 
+\begin_inset Formula $R=\infty$
+\end_inset
+
+, y si 
+\begin_inset Formula $\limsup_{n}\sqrt[n]{|a_{n}|}=\infty$
+\end_inset
+
+ se toma 
+\begin_inset Formula $R=0$
+\end_inset
+
+.
+ Por el criterio de la raíz, o el del cociente, la serie converge sólo en
+ la bola abierta 
+\begin_inset Formula $B(z_{0};R)$
+\end_inset
+
+, llamada 
+\series bold
+disco de convergencia
+\series default
+\SpecialChar endofsentence
+
+\end_layout
+
+\begin_layout Standard
+La serie de funciones 
+\begin_inset Formula $\sum_{n=0}^{\infty}f_{n}$
+\end_inset
+
+ 
+\series bold
+converge uniformemente
+\series default
+ en un conjunto 
+\begin_inset Formula $A$
+\end_inset
+
+ a una función 
+\begin_inset Formula $f$
+\end_inset
+
+ si 
+\begin_inset Formula $\forall\varepsilon>0,\exists n_{0}\in\mathbb{N}:\forall z\in A,m\geq n_{0};\left|f(z)-\sum_{n=0}^{m}f_{n}(z)\right|<\varepsilon$
+\end_inset
+
+.
+ El 
+\series bold
+criterio de Cauchy de convergencia uniforme
+\series default
+ afirma que una serie de funciones es uniformemente convergente en 
+\begin_inset Formula $A$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\forall\varepsilon>0,\exists n_{0}\in\mathbb{N}:\forall z\in A,n_{0}<p\leq q;\left|\sum_{n=p}^{q}f_{n}(z)\right|<\varepsilon$
+\end_inset
+
+, y el 
+\series bold
+criterio de Weierstrass
+\series default
+ afirma que si existe una serie de términos positivos 
+\begin_inset Formula $\sum_{n}b_{n}$
+\end_inset
+
+ convergente con 
+\begin_inset Formula $|f_{n}(z)|\leq b_{n}\forall z\in A,n\in\mathbb{N}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\sum_{n=0}^{\infty}f_{n}$
+\end_inset
+
+ converge uniformemente en 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La serie de potencias 
+\begin_inset Formula $\sum_{n}a_{n}(z-z_{0})^{n}$
+\end_inset
+
+ con radio de convergencia 
+\begin_inset Formula $R$
+\end_inset
+
+ converge absoluta y uniformemente en la bola cerrada 
+\begin_inset Formula $B[z_{0};r]$
+\end_inset
+
+ para cada 
+\begin_inset Formula $r<R$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\sum_{n}f_{n}$
+\end_inset
+
+ converge uniformemente en 
+\begin_inset Formula $A$
+\end_inset
+
+ y las 
+\begin_inset Formula $f_{n}$
+\end_inset
+
+ son continuas en 
+\begin_inset Formula $A$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+criterio de Abel
+\series default
+ afirma que, dada una serie de potencias 
+\begin_inset Formula $\sum_{n}a_{n}z^{n}$
+\end_inset
+
+, si para 
+\begin_inset Formula $z=c$
+\end_inset
+
+ la serie converge, también converge uniformemente en 
+\begin_inset Formula $[0,c]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $f(z):=\sum_{n=0}^{\infty}a_{n}z^{n}$
+\end_inset
+
+ para 
+\begin_inset Formula $z\in B(0;R)$
+\end_inset
+
+, siendo 
+\begin_inset Formula $R$
+\end_inset
+
+ el radio de convergencia de la serie, entonces la serie 
+\begin_inset Formula $\sum_{n=1}^{\infty}na_{n}z^{n-1}$
+\end_inset
+
+, obtenida derivando formalmente la anterior, tiene radio de convergencia
+ 
+\begin_inset Formula $R$
+\end_inset
+
+, y de hecho esta serie converge a la derivada de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es infinitamente derivable en el disco de convergencia y 
+\begin_inset Formula $a_{n}=\frac{f^{(n)}(0)}{n!}$
+\end_inset
+
+ para 
+\begin_inset Formula $n\geq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$
+\end_inset
+
+ con radio de convergencia 
+\begin_inset Formula $R$
+\end_inset
+
+, entonces la función 
+\begin_inset Formula $F$
+\end_inset
+
+ dada por 
+\begin_inset Formula $F(z):=\sum_{n=0}^{\infty}\frac{1}{n+1}a_{n}z^{n+1}$
+\end_inset
+
+ tiene radio de convergencia 
+\begin_inset Formula $R$
+\end_inset
+
+ y es primitiva de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Funciones elementales
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+exponencial compleja
+\series default
+ se define como
+\begin_inset Formula 
+\[
+e^{z}:=\sum_{n=0}^{\infty}\frac{1}{n!}z^{n}
+\]
+
+\end_inset
+
+Podemos ver que su radio de convergencia es infinito, 
+\begin_inset Formula $(e^{z})'=e^{z}$
+\end_inset
+
+ y 
+\begin_inset Formula $e^{z}e^{w}=e^{z+w}$
+\end_inset
+
+.
+ Además, 
+\begin_inset Formula $e^{x}>0$
+\end_inset
+
+ para todo 
+\begin_inset Formula $x\in\mathbb{R}$
+\end_inset
+
+, y es estrictamente creciente con
+\begin_inset Formula 
+\begin{eqnarray*}
+\lim_{x\rightarrow\infty}e^{x}=+\infty & \text{ y } & \lim_{x\rightarrow-\infty}e^{x}=0
+\end{eqnarray*}
+
+\end_inset
+
+ Definimos el 
+\series bold
+seno
+\series default
+ y el 
+\series bold
+coseno
+\series default
+, respectivamente, como
+\begin_inset Formula 
+\begin{eqnarray*}
+\sin x:=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!} & \text{ y } & \cos x:=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n}}{(2n)!}
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vemos que 
+\begin_inset Formula $e^{x+iy}=e^{x}e^{iy}=e^{x}(\cos y+i\sin y)$
+\end_inset
+
+, luego 
+\begin_inset Formula $\sin x=\text{Im}e^{ix}$
+\end_inset
+
+ y 
+\begin_inset Formula $\cos x=\text{Re}e^{ix}$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $|e^{iy}|^{2}=1$
+\end_inset
+
+, se tiene 
+\begin_inset Formula $\sin^{2}x+\cos^{2}x=1$
+\end_inset
+
+.
+ Además:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\sin'x=\cos x$
+\end_inset
+
+ y 
+\begin_inset Formula $\cos'x=-\sin x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\sin(-x)=-\sin x$
+\end_inset
+
+ y 
+\begin_inset Formula $\cos(-x)=\cos x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\sin(x+y)=\sin x\cos y+\cos x\sin y$
+\end_inset
+
+ y 
+\begin_inset Formula $\cos(x+y)=\cos x\cos y-\sin x\sin y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El conjunto 
+\begin_inset Formula $\{x>0:\cos x=0\}$
+\end_inset
+
+ es no vacío y de hecho tiene un primer elemento, que se denota 
+\begin_inset Formula $\frac{\pi}{2}$
+\end_inset
+
+.
+ Además, las funciones seno y coseno son 
+\begin_inset Formula $2\pi$
+\end_inset
+
+-periódicas, y 
+\begin_inset Formula $\psi:[0,2\pi)\rightarrow S$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\psi(t)=e^{it}$
+\end_inset
+
+ es una biyección de 
+\begin_inset Formula $[0,2\pi)$
+\end_inset
+
+ sobre la circunferencia unidad 
+\begin_inset Formula $S\subseteq\mathbb{C}$
+\end_inset
+
+.
+ Tenemos 
+\begin_inset Formula $\sin0=0$
+\end_inset
+
+, 
+\begin_inset Formula $\sin\frac{\pi}{2}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\sin\frac{\pi}{6}=\frac{1}{2}$
+\end_inset
+
+, 
+\begin_inset Formula $\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}$
+\end_inset
+
+, 
+\begin_inset Formula $\sin\frac{\pi}{3}=\frac{1}{\sqrt{3}}$
+\end_inset
+
+, 
+\begin_inset Formula $\sin t=\cos\left(\frac{\pi}{2}-t\right)$
+\end_inset
+
+ y 
+\begin_inset Formula $\cos t=\sin\left(\frac{\pi}{2}-t\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Por la biyección 
+\begin_inset Formula $\psi$
+\end_inset
+
+, y como dado 
+\begin_inset Formula $z\in\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{z}{|z|}\in S$
+\end_inset
+
+, existe un único 
+\begin_inset Formula $t\in[0,2\pi)$
+\end_inset
+
+, llamado 
+\series bold
+argumento principal
+\series default
+ de 
+\begin_inset Formula $z$
+\end_inset
+
+, tal que 
+\begin_inset Formula $z=|z|(\cos t+i\sin t)=|z|e^{it}$
+\end_inset
+
+.
+ Entonces:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $z_{1}z_{2}=|z_{1}|e^{it_{1}}|z_{2}|e^{it_{2}}=|z_{1}||z_{2}|e^{i(t_{1}+t_{2})}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\frac{1}{z}=z^{-1}=|z|^{-1}e^{-it}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $z^{n}=|z|^{n}e^{int}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Los 
+\begin_inset Formula $n$
+\end_inset
+
+ complejos de la forma 
+\begin_inset Formula $w=\sqrt[n]{|z|}e^{i\frac{2k\pi+t}{n}}$
+\end_inset
+
+ con 
+\begin_inset Formula $k=0,\dots,n-1$
+\end_inset
+
+ son los únicos con 
+\begin_inset Formula $w^{n}=z$
+\end_inset
+
+ para 
+\begin_inset Formula $z=|z|e^{it}$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document