Cauchy-Goursat Theorem/Examples/z squared over e to the it

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:

$\ds \oint_\gamma \map f z \rd z = 0$

$\ds \oint_\gamma \map f z \rd z$

$=$

$\ds \int_0^{2 \pi} i e^{i t} e^{2 i t} \rd t$

$\ds $

$=$

$\ds i \int_0^{2 \pi} e^{3 i t} \rd t$

$\ds $

$=$

$\ds \frac i {3 i} \paren {e^{6 i \pi} - e^{i 0} }$

$\ds $

$=$

$\ds \frac 1 3 \paren {\cos 6 \pi + i \sin 6 \pi - \cos 0 - i \sin 0}$

$\ds $

$=$

$\ds \frac 1 3 \paren {1 - 1}$

$\ds $

$=$

$\ds 0$

$\blacksquare$