This exercise gives an alternative proof of Cauchy's Integral Formula (Theorem 4.4.5) that does not depend on Cauchy's Theorem (Theorem 4.3.4). Suppose the region G is convex; this means that, whenever z and w are in G, the line segment between them is also in G. Suppose f is holomorphic in G, f′ is continuous, and γ is a positively oriented, simple, closed, piecewise smooth path, such that w is inside γ and γ∼G0.
(a) Consider the function g:[0,1]→C given by
g(t):=∫γz−wf(w+t(z−w))dz.
Show that g′=0.
Hint. Use Theorem A.0.9 (Leibniz's rule) and then find an antiderivative for ∂t∂f(w+t(z−w)).
(b) Prove Theorem 4.4.5 by evaluating g(0) and g(1).
(c) Why did we assume G is convex?