Cauchy-Goursat theorem

In complex analysis, the Cauchy-Goursat theorem supposes that if a complex-valued function $f(z)$ is analytic in a simply connected region $D$, then for any simple closed curve $C\in D$:

$${\oint }_{C}f(z)dz=0$$

which is provable using Green’s theorem, Stokes’ theorem, and the divergence theorem.

If $f$ is analytic everywhere within the region, then the vector field has zero curl and zero divergence, because they satisfy the Cauchy-Riemann equations.

Multiply connected regions

For multiply connected regions, let $c_1,\dots ,c_n$ be smaller contours within a larger contour $c$. If $f$ is analytic in the region inside $c$ and outside of $c_1,\dots ,c_n$, then it’s not true in general that ${\oint }_{C}f(z)dz=0$. Instead:

$${\oint }_{C}f(z)dz={\oint }_{c_1}f(z)dz+\cdots +{\oint }_{c_n}f(z)dz$$

Where there may be holes in the region, we use a method called deformation of contours, where we essentially take a slice between the outer and inner contours to get a single continuous contour that doesn’t enclose the holes.

We can assume a contour around the hole in the form of a circle (i.e., we parameterise as a circle and so on). This allows us to essentially evaluate an extremely complicated function in terms of circle contour integrals, which massively simplifies things for us.