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 $\triangle$ is a triangle contained completely inside $U$ (i.e. the interior and boundary are contained inside $U$). Then, we have:

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

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.

Statement in terms of cohomology and loops

Note that integration along piecewise smooth curves gives a map as follows:

Piecewise smooth curves $\times$ Continuous functions $\to \mathbb {C}$

We can restrict attention to piecewise smooth loops at a point $z_{0}\in U$. What this says is that if the loop is nullhomologous, or if we take a sum of loops whose homology classes add up to zero, then evaluating on any holomorphic function, gives the value zero.

This helps us tells us that the map we have descends to a bilinear map:

$H_{1}(U)\times H_{DR}^{1}(U)\to \mathbb {C}$

Facts used

• Cauchy-Riemann differential equations
• Green's formula