# Helicoid and Catenoid

Let $X$ and $Y$ be isothermal parametrizations of minimal surfaces such that their component functions are pairwise harmonic conjugates, then $X$ and $Y$ are called conjugate minimal surfaces.

My question is: Are the helicoid and the catenoid conjugate minimal surfaces? It seems to be impossible after a short calculation.

-

I remember running into calculation errors when I first did this problem, too. The trick that worked for me was to rotate the helicoid by an angle of $\frac{\pi}{2}$. Hopefully you should still have isothermal coordinates (check this), but now the Cauchy-Riemann Equations will be satisfied (check this too).
One way to look at this $\pi / 2$ rotation is as a complex rotation in the Weierstrass parametrization. –  Willie Wong Apr 6 '11 at 16:17