What is the harmonic conjugate of $v(x,y)=e^x$

Question

Letting $v(x,y)=e^x$, what is the complex conjugate of $v(x,y)=e^x$? i.e. find $u(x,y)$.

When I differentiate $e^x$ with respect to x and y, and use the Cauchy-Riemann equations, it seems impossible to then integrate either $dv/dx$ or $dv/dy$ to get the harmonic conjugate.

Answer 1

$u_{xx}+u_{yy}=e^x\neq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.