1. ## Removable Singularity

Having trouble with this question:

Let f be analytic at a point w

Define $g(z) = \frac{f(z)+wf'(w)-zf'(w)-f(w)}{(z-w)^2}$

Show that g has a removable singularity at w.

2. Just show that the numerator has a zero of order at least 2 at $w$ (can you see why this is what you have to do?).
4. Yes, exactly; show that the numerator is zero at $w$, and its derivative also.
The idea is that these zeroes will cancel the zeroes of the denominator - if you only had a simple zero or no zero at all at $w$, you'd end up with a simple or a double pole there.