 Author Topic: lec5101 Quiz3  (Read 397 times)

Wang Jingyao

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« on: October 09, 2020, 02:42:03 PM »
Let $\gamma_1$ be the semicircle from $1$ to $-1$ through $i$ and $\gamma_2$ the semicirlce from $1$ to $-1$ through $-i$. Compute $\int_{\gamma_1}z^2dz$ and $\int_{\gamma_2}z^2dz$. Can you account for the fact that they are equal?

Compute $\int_{\gamma_1}z^2dz$
\begin{align}
\gamma_1(t)=e^{it}\ for\ -\pi \leq t \leq 0\\
\notag \\
\gamma_1'(t)=ie^{it}\\
\notag \\
\int_{\gamma_1}z^2dz = \int^{\pi}_{0}(e^{it})^2 \cdot ie^{it}dt =  \dfrac{1}{3}e^{3it} \Big|^{\pi}_{0}=-\dfrac{2}{3}\\

\end{align}

Compute $\int_{\gamma_2}z^2dz$
\begin{align}
\gamma_2(t)=e^{it}\ for\ 0 \leq t \leq \pi\\
\notag \\
\gamma_2'(t)=ie^{it}\\
\notag \\
\int_{\gamma_2}z^2dz = \int^{\pi}_{0}(e^{it})^2 \cdot ie^{it}dt =  \dfrac{1}{3}e^{3it} \Big|^{\pi}_{0}=-\dfrac{2}{3}\\

\end{align}

\begin{align}
\gamma_1 + (-\gamma_2) = \gamma\\
\notag \\
\int_{\gamma}z^2dz = \int_{\gamma_1}z^2dz - \int_{\gamma_2}z^2dz =0\\
\notag \\
\therefore\ equal.
\end{align}