Basic question about complex function representation u(x,y)+iv(x,y) I suppose it natural to write $f(x,y) = u(x,y) + iv(x,y)$ but it make me wonder are we losing anything in that process? When we talk about things component-wise, is that limiting anything?
Can we always assume that given any complex function $f(x,y)$ that it could be written $u(x,y)+iv(x,y)$? (yes, I think)
Can we always assume that we can find formulas/equations for $u$ and $v$? (no, I think)
Are there situations where even though we are given the equation for $f$, it is not possible to find equations for u and v - more substantial ones than $u=Re(f)$ and $v=Im(f)$ obviously ? (I'm not sure about this one but I suspect that it can never happen but then again...)
Anyone with anything to add is appreciated.

If you have a formula for $\mathrm{f}$ then you are guaranteed a formula for $\mathrm{u}$ and $\mathrm{v}$. 
The real and imaginary parts of any complex number, and hence any complex valued function, can be found using the simple formulae
$$\Re(z) = \frac{1}{2}\left(z+\overline{z}\right)$$
$$\Im(z) = \frac{1}{2\mathrm{i}}\left(z-\overline{z}\right)$$
The formulae can be expressed in terms of $x$ any $y$ by making the substitution $z=x+y\mathrm{i}$ and doing lots of unsightly algebra.
A more interesting question is this: If I give you $\mathrm{u}(x,y)$ and $\mathrm{v}(x,y)$, can you find $\mathrm{f}(z)$?
