Let f(z) be analytic in a region G and set φ(z,w) = (f(w)-f(z))/(w-z) for w,z Є G w ≠ z. Let z0 Є G. Show that lim (z,w)-->(z0,z0) φ(z,w) =f'(z0).
Complex Variables. See attached file for full problem description
A proof needs to use some properties of ANALYTIC function.
You must have some definitions of an analytic function in your textbook(s) or lecture notes, but just in case here is a link to a web page with a definition:
The proof that you have already received is probably based on either the infinite differentiability of an analytic function or on its expandability in Taylor series,
f(t) = f(z0) + f'(z0)(t-z0) ...
Analytic functions and limits are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.