TITLE: Every harmonic function is the real part of a holomorphic function
QUESTION [12 upvotes]: Is there a way to show that every harmonic function is the real part of a holomorphic function without using integration equations if later theorems are allowed also?

REPLY [3 votes]: The statement is false without assumptions on the domain; log$| z |$ is harmonic on the punctured plane and can locally be expressed as the real part of an analytic function, but can't be the real part of an analytic function on any annulus about $0$ since the argument can't even be continuous.
As an aside, "annulus" gets spell-checked on this site!  haha