Complex Analysis - Maths KU

APP-6 Appendix II Proof of the Cauchy-Goursat Theorem
∆2, ... . Let z0 denote that point. Since the function f is analyticat z = z0,
f ′ (z0) exists. If we define
η(z) =
f(z) − f(z0)
z − z0
− f ′ (z0), (6)
then |η(z)| can be made arbitrarily small whenever z is sufficiently close to
z0. This fact—which will be used shortly—follows from the hypothesis that
f(z) − f(z0)
f is analyticin D and so the limit of the difference quotient as
z − z0
z → z0 exists and equals f ′ (z0). In the ε-δ symbolism of Definition 2.8, for
every ε>0, there exists a δ>0 such that
|η(z)|