Here is the problem:
If f= u+ iv is analytic in a region, show that uv is harmonic in the region but that u^2 need not be harmonic.
Thanks
For the complex analyst the definition is: a function said to be analytic at if its derivative exists at each point in some neighborhood of .
Moreover, a function is said to be harmonic if , that is the Laplace equation as you have noted.
Theorem: If is analytic is the domain , then both and are harmonic in .