Cauchy integral theorem

If a function is holomorphic on a simply connected domain, its integral around every closed piecewise smooth contour in that domain vanishes. In particular, contours with common endpoints may be deformed through the domain without changing the integral.

A holomorphic function on an open neighbourhood of a closed triangle has zero contour integral around its boundary. Goursat's subdivision proof uses complex differentiability alone: repeatedly choose a quarter-size triangle carrying at least one quarter of the integral, and subtract the affine expansion at the limiting point. No continuity of the derivative is assumed.

A contour deformation changes the path of a contour integral through a region where its integrand is a holomorphic function. The Cauchy integral theorem preserves the integral when the endpoints are fixed and the connecting boundary terms vanish. Crossing a pole instead contributes the appropriately oriented residue; a branch cut limits which deformations are permitted on one analytic sheet.

For $a>0$ and real $c$, integrate $e^{-a{z}^2}$ around a wide rectangle between the real axis and $\mathrm{Im}z=c$. The vertical contributions vanish as the width tends to infinity, so