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 1. If $f(z)$ is analytic in a domain $D$ and $D$ is simply connected, then $\int_\Gamma f(z) \,dz$ is zero for every closed loop $\Gamma$ lying within $D$.
 2. If $f$ is analytic in a domain $D$ then $\int_\Gamma f(z) \,dz = 0$ for any closed loop lying in $D$.
 3. If $f(z) = \frac{\,d}{\,dz}g(z)$ on a domain $D$ $($ $g(z)$ is analytic in $D$$)$ then the integral $\int_\Gamma f(z)\,dz$ is zero for every closed path $\Gamma$ lying within $D$.
 4. If $\int_\Gamma f(z) \,dz$ is zero for every closed path $\Gamma$ lying within $D$ then $f(z) = \frac{\,d}{\,dz}g(z)$ for some anlalytic function $g(z)$ and defined on all of $D$.
 5. If $f(z)$is analytic in a domain $D$$($ $D$ is not necessarily simply connected$)$ and the loop $\Gamma_1$ can be deformed to the loop $\Gamma_2$ within $D$, then $Re\left( \int_{\Gamma_1} f(z)\,dz\right) = Re\left( \int_{\Gamma_2} f(z) \,dz \right)$