# Complex analysis, harmonic conjugate determined up to a constant?

My question relates to the following problem: Given u(x,y) and that it is the real part of some analytic function f(z), find f(z).

So one way is to use the Cauchy-Riemann equations to build a harmonic conjugate v(x,y), which will be determined up to a constant. However, sometimes it seems easier to "guess" what f(z) is. The question is thus: Given that i've guessed some function g(z) that satisfies that u is its real part, how do i "show" that all analytic functions that solve the problem are can be written on the form g(z)+ic (if this is indeed true!)? It seems obvious that any function g(z)+ic does indeed solve the problem, but i want to motivate that this includes all the solutions.

Thank you for taking your time.

If I have understood your question correctly, $f-g$ takes values only on the imaginary axis. But a holomorphic map is open unless it is a constant on connected components. And $f-g$ is not open.