Inline Math Expression in Slides

The condition that U be simply connected means that U has no "holes" or, in homotopy terms, that the fundamental group of U is trivial; for instance, every open disk ${\displaystyle U=\{z:|z-z_{0}|<r\}}$ qualifies. The condition is crucial; consider

Block Math Expression in Slides

${\displaystyle \gamma (t)=e^{it}\quad t\in \left[0,2\pi \right]}$

which traces out the unit circle, and then the path integral

${\displaystyle \oint _{\gamma }{\frac {1}{z}}\,dz=\int _{0}^{2\pi }{ie^{it} \over e^{it}}\,dt=\int _{0}^{2\pi }i\,dt=2\pi i}$

is non-zero; the Cauchy integral theorem does not apply here since ${\displaystyle f(z)=1/z}$ because the domain of ${\displaystyle f}$ is not a convex set (${\displaystyle z=0)}$.