Suppose $f(z) = z^2$
This function has the component functions $u(x,y) = x^2 - y^2$, $v(x,y) = 2xy$
And it says in a book I'm reading that $v$ is a harmonic conjugate of $u$. But v is not harmonic as
$v_{xx} = 2$ and $v_{yy} = 2$
So $v_{xx} + v_{yy} = 4 \not= 0$
So how can it say that v is a harmonic conjugate of u? I presume I'm missing something as I don't think the book is wrong.