Goursat's integral lemma

Statement

For a triangle

Suppose $U$ is a domain in $\mathbb{C}$, and $f:U\to \mathbb{C}$ is a holomorphic function. Suppose $\mathrm{△}$ is a triangle contained completely inside $U$ (i.e. the interior and boundary are contained inside $U$). Then, we have:

${\int }_{\partial \mathrm{\Delta }}f(z)dz=0$

For a region bounded by piecewise smooth curves

Suppose $U$ is a domain in $\mathbb{C}$, and $f:U\to \mathbb{C}$ is a holomorphic function. Suppose $V$ is an open subset whose closure is a compact subset of $U$, such that $\partial V$ is piecewise $C^1$. Note that $V$ may have disconnected boundary; for instance, $V$ may be an annulus.

Then we have:

${\int }_{\partial V}f(z)dz=0$

Note that this is a slight generalization of the previous case, where we restrict $V$ to the interior of a triangle.

Facts used

• Cauchy-Riemann differential equations
• Green's formula