How do I:
Prove that if u and v are harmonic in a domain D, then (du/dy - dv/dx) + i(du/dx + dv/dy) is analytic in D.
A complex function is analytic if it obeys the Cauchy-Riemann equations. Apply those to the complex function you want to prove is analytic, and use the fact that $\displaystyle \frac{\partial^2u}{\partial{x}^2}+\frac{\partial^2 u}{\partial{y}^2}=0$ and $\displaystyle \frac{\partial^2v}{\partial{x}^2}+\frac{\partial^2 v}{\partial{y}^2}=0$, as the functions are harmonic.
--Kevin.