Answer to Question #142498 in Complex Analysis for Doll

Question

Prove that u(x,y) given by the following is harmonic obtain it's corresponding conjugate and original function f(z)
u(x,y)=e^xCosy

Answer 1

$u_x= e^x cos y=v_y.$ Hence $v=\int e^x cos\ y \ dy = e^x sin \ y + \phi(x)$. Hence $v_x= e^x sin y+\phi^{'}(x)=-u_y= e^x sin \ y.$ $\Rightarrow \phi^{'}(x)=0\Rightarrow \phi(x)=c$ , constant.

Hence $v(x,y)= e^x sin \ y+c.$ Hence $f(z)=u+iv=e^x(cos\ y+i \ sin \ y)+ic=e^{x+iy}+ic=e^z+ic$ .