Cauchy-Goursat Theorem/Example

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Example of use of Cauchy-Goursat Theorem

Let $\map \gamma t = e^{i t}$.

Let the domain of $\gamma$ be $\hointr 0 {2 \pi}$.

Let $\map f z = z^2$.


Then:

\(\displaystyle \oint_\gamma \map f z \rd z\) \(=\) \(\displaystyle \int_0^{2 \pi} i e^{i t} e^{2 i t} \rd t\)
\(\displaystyle \) \(=\) \(\displaystyle i \int_0^{2 \pi} e^{3 i t} \rd t\)
\(\displaystyle \) \(=\) \(\displaystyle \frac i {3 i} \paren {e^{6 i \pi} - e^{i 0} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 3 \paren {\cos 6 \pi + i \sin 6 \pi - \cos 0 - i \sin 0}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 3 \paren {1 - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle 0\)

$\blacksquare$