What is the general form of an analytic function with ux = 0 on a given domain?

[handiman]

In the question below I do not understand what is meant by the "general form".

'Suppose f(z) = u(x,y) + i*v(x,y) is analytic on a domain D, and u_x = 0 on D.
Find the general form of f(z).

 

[mathwonk]

maybe they mean find all possible such f. e.g y -ix seems to work, but there many others.