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.

# posted by rot13(Unafba Pune) @ 7:22 AM