Consider a piecewise smooth contour $\displaystyle \Gamma$ which is a triangle with vertices at $\displaystyle z = 0, 1 + i, i$. Evaluate the closed contour integral,

$\displaystyle $\int_\Gamma\bar{z}^3 dz$

around the triangle $\displaystyle \Gamma$ by suitably parametrizing the segments of the piecewise smooth contour $\displaystyle \Gamma$.

Why doesn’t Cauchy’s theorem apply to this integral?