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Complex Integral Cheat Sheet

The Winding Number

Let $\gamma$ be a Jordan curve in $\Omega$, and point $a \in \Omega$ not in $\gamma$. We define the winding number of $a$ with respect to $\gamma$ as:

\[n(\gamma, a) = \frac{1}{2\pi i} \int_\gamma \frac{dz}{z - a}\]

Cauchy’s Integral Theorem

Let $f(z)$ be holomorphic in a simply connected region $\Omega$, and a Jordan curve $\gamma$. Then the Cauchy’s integral theorem states:

\[\int_{\gamma} f(z) dz = 0\]

Cauchy’s Integral Formula

Let $f(z)$ be holomorphic in a simply connected region $\Omega$, and a Jordan curve $\gamma$ in $\Omega$ and a point $z \in \Omega$ not on $\gamma$ and such that $n(\gamma, z) = 1$ ($\gamma$ winds around $z$ exactly once, counter-clockwise). Then the Cauchy Integral formula is:

\[f(z) = \frac{1}{2\pi i} \int_{\gamma} \frac{f(w)}{w - z} dw\]