(a) Show that an analytic function f(z) defined in a simply connected domain Ω is constant if R(f(z)) (= the real part of f(z)) is constant throughout Ω.
(b) Let f(z) be analytic and non-vanishing in a domain Ω. Show that ln l f(z) l is a harmonic function in Ω.
"Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics" 3rd Edition, Saff and Snider, Prentice Hall.
This solution goes through a complex analysis problem which pertains to analytic and harmonic functions.