What's the intuition behind $f'(z)= \frac{\partial f}{\partial x}$ and $f'(z)= \frac{1}{i} \frac{\partial f}{\partial y}$, where $f : \mathbb{C} \to \mathbb{C}$ is assumed to be analytic?
I know the algebraic proof of this, but I am not sure what these equations really mean. If anyone could explain why these equations should be true, I would be grateful!
EDIT : Having thought about it a bit more I think these equations are not so dramatic. For an analytic function the derivative must agree with the directional derivatives ( this can probably be said better ) and this is, I suppose, just what these equations are saying.
$f'(z) = \lim_{h \to 0} {f(z+h)-f(z) \over h }$, where $h \in \mathbb{C}$.
Hence $f'(z) = \lim_{t \to 0, t \in \mathbb{R}} {f(z+t)-f(z) \over t } = \partial_x f(z)$ and $f'(z) = \lim_{t \to 0, t \in \mathbb{R}} {f(z+it)-f(z) \over it } = {1 \over i}\partial_y f(z)$.