Having trouble with this question:
Let f be analytic at a point w
Define $\displaystyle g(z) = \frac{f(z)+wf'(w)-zf'(w)-f(w)}{(z-w)^2}$
Show that g has a removable singularity at w.
Please help!
Yes, exactly; show that the numerator is zero at $\displaystyle w$, 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 $\displaystyle w$, you'd end up with a simple or a double pole there.