$(Existenceof$ conjugate harmonic function$)$ Let $u(x,y)be$

a given function that $is$ harmonic $in$ the rectangle $x_{1}x$

$x_2,y_{1}y{y}_2.$ Prove that there exists a conjugate harmonic

function $v(x,y)in$ the rectangle $Df=u+$ivDv$(x,y)del\frac{u}{d}$elx$=$del$\frac{v}{d}$ely

$u(x,y)={\displaystyle \underset{y0}{\overset{y}{}}}\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{x}}(x,y^{\text{'}})del{y}^{\text{'}}+A(x)uA(x)=-{\displaystyle \underset{x_0}{\overset{x}{}}}\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}(x^{\text{'}},y_0)d{x}^{\text{'}}{x}_0,y_{0}Dv(x,y)={\displaystyle \underset{y_0}{\overset{y}{}}}\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}(x.y^{\text{'}})del{y}^{\text{'}}-{\displaystyle \underset{x_0}{\overset{x}{}}}\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}(x^{\text{'}},y_0)d{x}^{\text{'}}Dv(x,y)={\displaystyle \underset{(x{)}^{(x,y)}}{\overset{\phantom{}}{}}},y,y,y$