# $\color{ffffff}\colorbox{#016b32}{- Ecuaciones de Cauchy-Riemann -}$ --- 🟢 #Teorema > [!VCTeorema] $~$ Ecuaciones de Cauchy-Riemann > > Sean $A\subseteq \mathbb{C}$ un [[Conjunto Abierto en C|abierto]] y $f:A\to \mathbb{C}$ una [[Función|función]] tal que $f(z):=u(x,y)+i~v(x,y)$. > > Si $f$ es [[Función Holomorfa|holomorfa]] en $z_{0}\in A$ con $z_{0}:=x_{0}+iy_{0}$, entonces se tiene que: > > $\frac{\partial u}{\partial x}(x_{0},y_{0})=\frac{\partial v}{\partial y}(x_{0},y_{0})~~~~\wedge~~~~\frac{\partial u}{\partial y}(x_{0},y_{0})=-\frac{\partial v}{\partial x}(x_{0},y_{0})$ > Además, > > $f'(z_{0})=\frac{\partial u}{\partial x}(x_{0},y_{0})+i~\frac{\partial v}{\partial x}(x_{0},y_{0})=\frac{\partial v}{\partial y}(x_{0},y_{0})-i~\frac{\partial u}{\partial y}(x_{0},y_{0})$ > No olvidemos que, $u:U\to \mathbb{R}$ y $v:U\to \mathbb{R}$ son [[Función|funciones]] con $U:=\{(x,y)\in \mathbb{R}^{2}\mid \exists~z\in A: z=x+iy\}$ --- ##### <font style="color:#14a356"> Demostración: </font> Dado que $f$ es [[Función Holomorfa|holomorfa]] en $z_{0}$, sabemos que: $f'(z)=\lim_{z\to z_{0}}\frac{f(z)-f(z_{0})}{z-z_{0}}=\lim_{z\to z_{0}}\frac{u(x,y)+i~v(x,y)- u(x_{0},y_{0})-i~v(x_{0},y_{0})}{(x-x_{0})+i~(y-y_{0})}$ En particular, si $z\to z_{0}$ a través de la [[rectas|recta]] $\{z\in \mathbb{C}\mid \Im(z)=y_{0}\}$, obtenemos que, $f'(z_{0})=\lim_{x\to x_{0}}\frac{u(x,y_{0})+i~v(x,y_{0})- u(x_{0},y_{0})-i~v(x_{0},y_{0})}{(x-x_{0})}$ $~~~~~~~~~~~~~~~~~~~=\lim_{x\to x_{0}}\frac{u(x,y_{0})- u(x_{0},y_{0})}{(x-x_{0})}+i~\lim_{x\to x_{0}}\frac{v(x,y_{0})- v(x_{0},y_{0})}{(x-x_{0})}$ $=\frac{\partial u}{\partial x}(x_{0},y_{0})+i\frac{\partial v}{\partial x}(x_{0},y_{0})$ Esto aplicando [[propiedades del límite complejo]] y recordando la definición de [[Derivada Parcial|derivada parcial]]. Análogamente, si $z\to z_{0}$ a través de la [[rectas|recta]] $\{z\in \mathbb{C}\mid \Re(z)=x_{0}\}$, $f'(z_{0})=\lim_{y\to y_{0}}\frac{u(x_{0},y)+i~v(x_{0},y)- u(x_{0},y_{0})-i~v(x_{0},y_{0})}{i~(y-y_{0})}$ $~~~~~~~~~~~~~~~~~~~=\lim_{y\to y_{0}}\frac{u(x_{0},y)- u(x_{0},y_{0})}{i~(y-y_{0})}+i~\lim_{y\to y_{0}}\frac{v(x_{0},y)- v(x_{0},y_{0})}{i~(y-y_{0})}$ $=\frac{\partial v}{\partial y}(x_{0},y_{0})-i\frac{\partial u}{\partial y}(x_{0},y_{0})$ Por lo tanto, igualando ambas expresiones de $\Re(f'(z_{0}))$ y $\Im(f'(z_{0}))$, $\frac{\partial u}{\partial x}(x_{0},y_{0})=\frac{\partial v}{\partial y}(x_{0},y_{0})~~~~\wedge~~~~\frac{\partial u}{\partial y}(x_{0},y_{0})=-\frac{\partial v}{\partial x}(x_{0},y_{0})$ Además, hemos alcanzado también dos igualdades para $f'(z_{0})$.<font style="color:#14a356"> $~~\square$ </font> --- #### <font style="color:#14a356"> Links:</font> [[Conjunto Abierto en C|abierto]] | [[Función|función]] ---