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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.

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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.

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  • $\begingroup$ Thank you, this is exactly what i was looking for. $\endgroup$
    – user256860
    Sep 24, 2016 at 17:19
  • $\begingroup$ Surely the word "connected" deserves mention somewhere. $\endgroup$
    – zhw.
    Sep 24, 2016 at 18:50

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