diff options
Diffstat (limited to 'fvc')
| -rw-r--r-- | fvc/n2.lyx | 44 | ||||
| -rw-r--r-- | fvc/n3.lyx | 10 | ||||
| -rw-r--r-- | fvc/n4.lyx | 50 |
3 files changed, 52 insertions, 52 deletions
@@ -91,7 +91,7 @@ Teorema de Cauchy-Goursat: \end_inset y -\begin_inset Formula $\Delta(a,b,c):=\{\mu a+\lambda b+\gamma c\mid \mu+\lambda+\gamma=1;\mu,\lambda,\gamma\geq0\}\subseteq\Omega$ +\begin_inset Formula $\Delta(a,b,c)\coloneqq \{\mu a+\lambda b+\gamma c\mid \mu+\lambda+\gamma=1;\mu,\lambda,\gamma\geq0\}\subseteq\Omega$ \end_inset , entonces @@ -111,23 +111,23 @@ Teorema de Cauchy-Goursat: Demostración: \series default Sean -\begin_inset Formula $\gamma:=[a,b,c,a]$ +\begin_inset Formula $\gamma\coloneqq [a,b,c,a]$ \end_inset , -\begin_inset Formula $\Delta:=\Delta(a,b,c)$ +\begin_inset Formula $\Delta\coloneqq \Delta(a,b,c)$ \end_inset , -\begin_inset Formula $a':=\frac{b+c}{2}$ +\begin_inset Formula $a'\coloneqq \frac{b+c}{2}$ \end_inset , -\begin_inset Formula $b':=\frac{a+c}{2}$ +\begin_inset Formula $b'\coloneqq \frac{a+c}{2}$ \end_inset , -\begin_inset Formula $c':=\frac{a+b}{2}$ +\begin_inset Formula $c'\coloneqq \frac{a+b}{2}$ \end_inset e @@ -156,7 +156,7 @@ Sean \begin_layout Itemize Si -\begin_inset Formula $|J_{k}|:=\max_{i}|J_{i}|$ +\begin_inset Formula $|J_{k}|\coloneqq \max_{i}|J_{i}|$ \end_inset , @@ -206,7 +206,7 @@ Para \end_inset , -\begin_inset Formula $F(x):=\frac{x+a}{2}$ +\begin_inset Formula $F(x)\coloneqq \frac{x+a}{2}$ \end_inset es una biyección de @@ -218,11 +218,11 @@ Para \end_inset , pues si -\begin_inset Formula $x:=ra+sb+tc$ +\begin_inset Formula $x\coloneqq ra+sb+tc$ \end_inset , -\begin_inset Formula $F(x):=\frac{ra+sb+tc+a}{2}=\frac{ra+sb+tc+(r+s+t)a}{2}=ra+s\frac{a+b}{2}+t\frac{a+c}{2}=ra+sc'+tb'$ +\begin_inset Formula $F(x)\coloneqq \frac{ra+sb+tc+a}{2}=\frac{ra+sb+tc+(r+s+t)a}{2}=ra+s\frac{a+b}{2}+t\frac{a+c}{2}=ra+sc'+tb'$ \end_inset . @@ -236,7 +236,7 @@ Para \end_inset la biyección -\begin_inset Formula $F(x):=\frac{a+b+c-x}{2}$ +\begin_inset Formula $F(x)\coloneqq \frac{a+b+c-x}{2}$ \end_inset . @@ -245,11 +245,11 @@ Para \end_deeper \begin_layout Standard Sean entonces -\begin_inset Formula $I_{1}:=\max_{i}|J_{i}|$ +\begin_inset Formula $I_{1}\coloneqq \max_{i}|J_{i}|$ \end_inset , -\begin_inset Formula $\gamma_{1}:=[a_{1},b_{1},c_{1},a_{1}]$ +\begin_inset Formula $\gamma_{1}\coloneqq [a_{1},b_{1},c_{1},a_{1}]$ \end_inset la curva correspondiente a @@ -257,7 +257,7 @@ Sean entonces \end_inset y -\begin_inset Formula $\Delta_{1}:=\Delta(a_{1},b_{1},c_{1})$ +\begin_inset Formula $\Delta_{1}\coloneqq \Delta(a_{1},b_{1},c_{1})$ \end_inset , con lo que @@ -297,7 +297,7 @@ Sean entonces . Sea -\begin_inset Formula $p(z):=f(\alpha)+f'(\alpha)(z-\alpha)$ +\begin_inset Formula $p(z)\coloneqq f(\alpha)+f'(\alpha)(z-\alpha)$ \end_inset una función polinómica y por tanto con primitiva, entonces @@ -539,11 +539,11 @@ Si \end_inset , sean -\begin_inset Formula $c_{\rho}:=(1-\rho)a+\rho b$ +\begin_inset Formula $c_{\rho}\coloneqq (1-\rho)a+\rho b$ \end_inset y -\begin_inset Formula $b_{\rho}:=(1-\rho)a+\rho c$ +\begin_inset Formula $b_{\rho}\coloneqq (1-\rho)a+\rho c$ \end_inset para @@ -1221,7 +1221,7 @@ Demostración: . Sea -\begin_inset Formula $f(z):=\frac{1}{p(z)}$ +\begin_inset Formula $f(z)\coloneqq \frac{1}{p(z)}$ \end_inset , @@ -1279,7 +1279,7 @@ Demostración: . Sea entonces -\begin_inset Formula $g(z):=\frac{1}{f(z)-\alpha}$ +\begin_inset Formula $g(z)\coloneqq \frac{1}{f(z)-\alpha}$ \end_inset una función entera, como @@ -1393,7 +1393,7 @@ luego \end_inset , por el teorema de Taylor, sea -\begin_inset Formula $c_{n}:=\frac{F^{(n)}(\alpha)}{n!}$ +\begin_inset Formula $c_{n}\coloneqq \frac{F^{(n)}(\alpha)}{n!}$ \end_inset , como @@ -1473,7 +1473,7 @@ Teorema de convergencia de Weierstrass: \end_inset y -\begin_inset Formula $f(z):=\lim_{n}f_{n}(z)$ +\begin_inset Formula $f(z)\coloneqq \lim_{n}f_{n}(z)$ \end_inset para @@ -1583,7 +1583,7 @@ Sean \end_inset y -\begin_inset Formula $H:=\{z\in\mathbb{C}\mid d(z,K)\leq\rho\}$ +\begin_inset Formula $H\coloneqq \{z\in\mathbb{C}\mid d(z,K)\leq\rho\}$ \end_inset , con lo que @@ -87,7 +87,7 @@ Sean \end_inset y -\begin_inset Formula $Z(f):=\{z\in\Omega\mid f(z)=0\}$ +\begin_inset Formula $Z(f)\coloneqq \{z\in\Omega\mid f(z)=0\}$ \end_inset , @@ -139,7 +139,7 @@ f(z)=\sum_{n=0}^{\infty}c_{n}(z-a)^{n} \end_inset para -\begin_inset Formula $c_{n}:=\frac{f^{(n)}(a)}{n!}$ +\begin_inset Formula $c_{n}\coloneqq \frac{f^{(n)}(a)}{n!}$ \end_inset , y queremos ver que todos los @@ -169,7 +169,7 @@ para \end_inset Sea -\begin_inset Formula $g_{k}(z):=\sum_{n=k+1}^{\infty}c_{n}(z-a)^{n-k}$ +\begin_inset Formula $g_{k}(z)\coloneqq \sum_{n=k+1}^{\infty}c_{n}(z-a)^{n-k}$ \end_inset una función holomorfa en @@ -210,7 +210,7 @@ status open \end_inset Sea -\begin_inset Formula $A:=\{z\in\Omega\mid \forall k\in\mathbb{N},f^{(k)}(z)=0\}\neq\emptyset$ +\begin_inset Formula $A\coloneqq \{z\in\Omega\mid \forall k\in\mathbb{N},f^{(k)}(z)=0\}\neq\emptyset$ \end_inset , pues @@ -337,7 +337,7 @@ principio de identidad para funciones holomorfas \end_inset no es idénticamente nula, entonces todo punto de -\begin_inset Formula $Z(f):=\{z\in\Omega\mid f(z)=0\}$ +\begin_inset Formula $Z(f)\coloneqq \{z\in\Omega\mid f(z)=0\}$ \end_inset es aislado y @@ -108,7 +108,7 @@ Toda curva Demostración: \series default Sean -\begin_inset Formula $\rho:=\min_{t\in[a,b]}|\gamma(t)|>0$ +\begin_inset Formula $\rho\coloneqq \min_{t\in[a,b]}|\gamma(t)|>0$ \end_inset , @@ -140,7 +140,7 @@ Demostración: \end_inset y -\begin_inset Formula $D_{k}:=D(\gamma(t_{k}),\rho)$ +\begin_inset Formula $D_{k}\coloneqq D(\gamma(t_{k}),\rho)$ \end_inset . @@ -180,11 +180,11 @@ Demostración: . Sean ahora -\begin_inset Formula $\theta_{k}(t):=A_{k}(\gamma(t))\in\text{Arg}(\gamma(t))$ +\begin_inset Formula $\theta_{k}(t)\coloneqq A_{k}(\gamma(t))\in\text{Arg}(\gamma(t))$ \end_inset y -\begin_inset Formula $m_{k}:=\theta_{k}(t_{k})-\theta_{k+1}(t_{k})$ +\begin_inset Formula $m_{k}\coloneqq \theta_{k}(t_{k})-\theta_{k+1}(t_{k})$ \end_inset , y definimos @@ -192,7 +192,7 @@ Demostración: \end_inset como -\begin_inset Formula $\theta(t):=\theta_{k}(t)+\sum_{i=0}^{k-1}m_{k}$ +\begin_inset Formula $\theta(t)\coloneqq \theta_{k}(t)+\sum_{i=0}^{k-1}m_{k}$ \end_inset para @@ -334,7 +334,7 @@ Sean \end_inset , -\begin_inset Formula $\rho:=\min_{t\in[a,b]}|\gamma(t)-z_{0}|>0$ +\begin_inset Formula $\rho\coloneqq \min_{t\in[a,b]}|\gamma(t)-z_{0}|>0$ \end_inset y @@ -371,7 +371,7 @@ Sean \end_inset , tenemos que -\begin_inset Formula $\theta(t):=\theta_{0}(t)+\arg\frac{\gamma(t)-z}{\gamma(t)-z_{0}}$ +\begin_inset Formula $\theta(t)\coloneqq \theta_{0}(t)+\arg\frac{\gamma(t)-z}{\gamma(t)-z_{0}}$ \end_inset es un argumento continuo de @@ -499,7 +499,7 @@ Demostración: \end_inset , entonces -\begin_inset Formula $\varphi(t):=\log|\gamma(t)-z|+i\theta(t)$ +\begin_inset Formula $\varphi(t)\coloneqq \log|\gamma(t)-z|+i\theta(t)$ \end_inset es un logaritmo continuo de @@ -525,7 +525,7 @@ Demostración: es derivable. Entonces -\begin_inset Formula $\varphi_{k}:=\varphi|_{[t_{k-1},t_{k}]}$ +\begin_inset Formula $\varphi_{k}\coloneqq \varphi|_{[t_{k-1},t_{k}]}$ \end_inset también lo es y @@ -560,7 +560,7 @@ Una cadena \series default es una expresión de la forma -\begin_inset Formula $\Gamma:=m_{1}\gamma_{1}+\dots+m_{q}\gamma_{q}$ +\begin_inset Formula $\Gamma\coloneqq m_{1}\gamma_{1}+\dots+m_{q}\gamma_{q}$ \end_inset donde los @@ -581,7 +581,7 @@ soporte \end_inset a -\begin_inset Formula $\Gamma^{*}:=\bigcup_{k}\gamma_{k}^{*}$ +\begin_inset Formula $\Gamma^{*}\coloneqq \bigcup_{k}\gamma_{k}^{*}$ \end_inset y @@ -593,16 +593,16 @@ longitud \end_inset a -\begin_inset Formula $\ell(\Gamma):=\sum_{k}|m_{k}|\ell(\gamma_{k})$ +\begin_inset Formula $\ell(\Gamma)\coloneqq \sum_{k}|m_{k}|\ell(\gamma_{k})$ \end_inset . Si -\begin_inset Formula $\Sigma:=n_{1}\sigma_{1}+\dots+n_{p}\sigma_{p}$ +\begin_inset Formula $\Sigma\coloneqq n_{1}\sigma_{1}+\dots+n_{p}\sigma_{p}$ \end_inset es otra cadena, llamamos -\begin_inset Formula $\Gamma+\Sigma:=m_{1}\gamma_{1}+\dots+m_{q}\gamma_{q}+k_{1}\sigma_{1}+\dots+k_{p}\sigma_{p}$ +\begin_inset Formula $\Gamma+\Sigma\coloneqq m_{1}\gamma_{1}+\dots+m_{q}\gamma_{q}+k_{1}\sigma_{1}+\dots+k_{p}\sigma_{p}$ \end_inset . @@ -647,7 +647,7 @@ ciclo \end_inset a -\begin_inset Formula $\text{Ind}_{\Gamma}(z):=\sum_{k}m_{k}\text{Ind}_{\gamma_{k}}(z)$ +\begin_inset Formula $\text{Ind}_{\Gamma}(z)\coloneqq \sum_{k}m_{k}\text{Ind}_{\gamma_{k}}(z)$ \end_inset . @@ -806,7 +806,7 @@ es continua en \end_inset Como -\begin_inset Formula $K:=\{\{z_{n}\}_{n}\cup\{a\}\}\times\Gamma^{*}$ +\begin_inset Formula $K\coloneqq \{\{z_{n}\}_{n}\cup\{a\}\}\times\Gamma^{*}$ \end_inset es compacto por ser producto de compactos, @@ -889,7 +889,7 @@ Si además, para \end_inset dada por -\begin_inset Formula $F_{w}(z):=F(z,w)$ +\begin_inset Formula $F_{w}(z)\coloneqq F(z,w)$ \end_inset es holomorfa en @@ -1059,7 +1059,7 @@ Ahora bien, fijado \end_inset , sea -\begin_inset Formula $F_{w}(z):=F(w,z)$ +\begin_inset Formula $F_{w}(z)\coloneqq F(w,z)$ \end_inset , es claro que @@ -1083,7 +1083,7 @@ Ahora bien, fijado \begin_layout Standard Sea -\begin_inset Formula $\Omega_{0}:=\{z\in\mathbb{C}\setminus\Gamma^{*}\mid \text{Ind}_{\Gamma}(z)=0\}$ +\begin_inset Formula $\Omega_{0}\coloneqq \{z\in\mathbb{C}\setminus\Gamma^{*}\mid \text{Ind}_{\Gamma}(z)=0\}$ \end_inset , que es abierto por ser unión de componentes conexas de @@ -1123,7 +1123,7 @@ status open \end_inset dada por -\begin_inset Formula $F_{0}(z,w):=\frac{f(w)}{w-z}$ +\begin_inset Formula $F_{0}(z,w)\coloneqq \frac{f(w)}{w-z}$ \end_inset . @@ -1302,7 +1302,7 @@ forma general del teorema de Cauchy \end_inset , aplicando la fórmula integral de Cauchy a -\begin_inset Formula $g(z):=(z-a)f(z)$ +\begin_inset Formula $g(z)\coloneqq (z-a)f(z)$ \end_inset , como @@ -1886,7 +1886,7 @@ status open . Sea -\begin_inset Formula $K:=\mathbb{C}\setminus\Omega_{0}=\Gamma^{*}\cup\{z\in\mathbb{C}\setminus\Gamma^{*}\mid \text{Ind}_{\Gamma}(z)\neq0\}$ +\begin_inset Formula $K\coloneqq \mathbb{C}\setminus\Omega_{0}=\Gamma^{*}\cup\{z\in\mathbb{C}\setminus\Gamma^{*}\mid \text{Ind}_{\Gamma}(z)\neq0\}$ \end_inset , que es cerrado por ser complementario de un abierto y acotado porque no @@ -1934,15 +1934,15 @@ Sean \end_inset , -\begin_inset Formula $m_{k}:=\text{Ind}_{\Gamma}(a_{k})$ +\begin_inset Formula $m_{k}\coloneqq \text{Ind}_{\Gamma}(a_{k})$ \end_inset , -\begin_inset Formula $\gamma_{k}:=C(a_{k},\rho)$ +\begin_inset Formula $\gamma_{k}\coloneqq C(a_{k},\rho)$ \end_inset y -\begin_inset Formula $\Sigma:=\sum_{k=1}^{q}m_{k}\gamma_{k}$ +\begin_inset Formula $\Sigma\coloneqq \sum_{k=1}^{q}m_{k}\gamma_{k}$ \end_inset . |