Might be a stupid question, but this has been bugging me. I'm revising my complex analysis course and I always come across writing $f(z)=u(x,y)+iv(x,y)$ to prove some theorem about complex functions, my question is: is this always possible? If yes, why? I can't think of a function for which this wouldn't be possible, but then I'm sure there are people who thought about it a lot more than me.

My idea is that this separation is always possible if the function is holomorphic at some point $z\_0$, because then it would be possible to expand it as a Laurent series about that point, and since $(z-z\_0)^k$ can be expanded in a sum of terms, one can then separate the real and imaginary part.

Does my reasoning hold? Is there a simpler way to think about it?

Thank you