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.