Use the Cauchy-Riemann equations to determine all differentiable
functions that satisfy $Re(f(z))=xy$
I think I know how to do this problem. If we let $z=x+iy$, then $f(z)=u(x,y)+iv(x,y)$. We are given $u(x,y)=xy$. The Cauchy-Riemann equations give us, after some calculations, that $f(x,y)=xy+i(\frac{y^2-x^2}{2}+C)$, $C \in \mathbb{R}$.
I'm somewhat unsatisfied that my answer is expressed in terms of $x$ and $y$; I'd like to have it in terms of one complex variable $z$ rather than two real variables $x$ and $y$ [even if they are equivalent].
I've tried doing this with the identities $x=\frac{z+\bar{z}}{2}$ and $y=\frac{z-\bar{z}}{2i}$, but I arrive at something involving $z$ and $\bar{z}$. It's my understanding that differentiable (and hence analytic) functions shouldn't have a $\bar{z}$ in their formulas, so what am I doing wrong?
Best Answer
What you have is $-\frac{i}{2} z^2 + iC$, and this seems fine. The issue ought to be in the step you did not detail.