A First Course in Complex Analysis · Problem 26

Consequences of Cauchy's Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchys-theorem

Problem

This exercise gives an alternative proof of Corollary 4.3.8 via Green's Theorem A.0.10. Suppose

G \subseteq \mathbb{C}

is a region,

f

is holomorphic in

G

f'

is continuous,

\gamma

is a simple piecewise smooth closed curve, and

\gamma \sim_G 0

. Explain that we may write

\int_{\gamma} f(z)\, dz = \int_{\gamma} (u + iv)(dx + i\, dy) = \int_{\gamma} u\, dx - v\, dy + i\int_{\gamma} v\, dx + u\, dy

and show that these integrals vanish, by using Green's Theorem A.0.10 together with Proposition 4.4.6, and then the Cauchy--Riemann equations (2.2).