Chapter 14.3, Problem 13P

In Chapter 6, Section 11, we showed that a necessary condition for ${\displaystyle {\int }_a^{b}F}\cdot dr$ to be independent of the path of integration, that is, for ${\displaystyle {\oint }_{C}F}\cdot dr$ around a simple closed curve $C$ to be zero, was curl $F=0,$ or in two dimensions, $\partial F_y/\partial x=\partial F_x/\partial y.$ By considering ( $3.2)$, show that the corresponding condition for ${\displaystyle {\oint }_{C}f}(z)dz$ to be zero is that the Cauchy-Riemann conditions hold.