Just a quick question that I hope nobody would mind answering. I'm reading through a page on this book http://books.google.com/books?id=dslVpWam-4QC&pg=PA252#v=onepage&q&f=false, and I'm a bit confused as to how the $U(\zeta)$ they defined in the middle of the page is harmonic for $u$ being a harmonic function. I've been trying to show that it's harmonic by showing that the Laplacian is zero, but I'm not having much luck after two pages of algebra.
The answer is probably pretty obvious, and I might just be forgetting something at this point, but I would greatly appreciate your help!
This function is the composition of a harmonic function with a holomorphic function. So it remains harmonic.