Cauchy's Theorem

Suppose \(D\) is a simply connected domain in \(\mathbb{C}\), \(\,f\) be analytic, and \(\gamma: [a, b] \mapsto D\) be a piecewise smooth, closed curve in \(D\) (so that \(\gamma(b) = \gamma(a)\)), then

\[ \begin{aligned} \oint_\gamma f(z)\,dz = 0 \\ \end{aligned} \]

!

Corollary

Suppose \(\gamma_1\) and \(\gamma_2\) are two simple closed curves (ie neither of them intersects itself), oriented clockwise, where \(\gamma_2\) is inside \(\gamma_1\). Suppose \(f\) is analytic in a domain \(D\) that contains both curves as well as the region between them, then

\[ \begin{aligned} \oint_{\gamma_1} f(z)\,dz = \oint_{\gamma_2} f(z)\,dz \\ \end{aligned} \]

!

For example, suppose \(R\) is the rectangle with vertices \(−2−i, 2−i, 2+i, −2+i\), the integral

\[ \begin{aligned} \oint_{\partial R} \frac{1}{z - z_0} \,dz \\ \end{aligned} \]

can have zero vs. non-zero value depending \(z_0\). Can you see why and how to compute the non-zero value ?

More at Analysis of a Complex Kind.