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**One last simple question about complex analysis....**

Hi, sorry again for having made so many threads. I have one remaining question about complex analysis that I keep get confused on.

Say that I have some complex function h(z). Sometimes I am really confused how to break that down into h(z) = u(x,y) + iv(x,y)

So for example say I want to take the integral ∫z

^{2}dz from 0 --> 1+i.

How do I know the u(x,y) and v(x,y) real functions that h(z) it is made up of? I mean... what if I want to see if the function is holomorphic? Or exact? Or harmonic? Or anything like that. I need the u and v components. For some functions, I just know it by a formula (like e

^{z}= e

^{x}cos(y)+ie

^{x}sin(y) but I am really at a loss when I am just handed any arbitrary function. how do I find it? In fact, I'm not even sure how the formula for the complex exponential function was found. Was it just by converting to polar coordinates? I'm very confused on this point