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**Quiz 4 / Quiz-5101-C**

« **on:**October 23, 2020, 03:48:21 PM »

$\textbf {Problem:} \\\\ $

$\text{Evaluate the given integral using the technique of Example 10 of Section 2.3:} \\$

$\begin{gather}

\int_{\gamma} e^{z}\, dz

\end{gather}$

$\text{where}\ \gamma \ \text{is the semicircle from -1 to 1 passing through i.}$

$\textbf{Solution: } \\\\$

$\text{We have integrand} f(z) = e^{z} \text{, and it is the derivative of }F(z)=e^{z} . \\\\$

$\text{This is valid when } F(z) \text{is analytic on domain D.}\\\\$

$\text{Indeed, both} F(z) \text{ and } f(z) \text{ is analytic on the semicircle.}\\\\$

$\text{Therefore, we have}\\\\$

$

\begin{gather}

\begin{aligned}

\int_{\gamma} e^{z}\, dz &{} = \int_{\gamma} f(z)\, dz \\\\

&{} = \int_{\gamma} F'(z)\, dz \\\\

&{} = \text{F(endpoint) \m F(initialpoint)} \\\\

&{} = F(1) - F(-1) \\\\

&{} = e - e^{-1}

\end{aligned}

\end{gather}$

$\text{Therefore} \ e - e^{-1} \ \text{is our final answer.}$

$\text{Evaluate the given integral using the technique of Example 10 of Section 2.3:} \\$

$\begin{gather}

\int_{\gamma} e^{z}\, dz

\end{gather}$

$\text{where}\ \gamma \ \text{is the semicircle from -1 to 1 passing through i.}$

$\textbf{Solution: } \\\\$

$\text{We have integrand} f(z) = e^{z} \text{, and it is the derivative of }F(z)=e^{z} . \\\\$

$\text{This is valid when } F(z) \text{is analytic on domain D.}\\\\$

$\text{Indeed, both} F(z) \text{ and } f(z) \text{ is analytic on the semicircle.}\\\\$

$\text{Therefore, we have}\\\\$

$

\begin{gather}

\begin{aligned}

\int_{\gamma} e^{z}\, dz &{} = \int_{\gamma} f(z)\, dz \\\\

&{} = \int_{\gamma} F'(z)\, dz \\\\

&{} = \text{F(endpoint) \m F(initialpoint)} \\\\

&{} = F(1) - F(-1) \\\\

&{} = e - e^{-1}

\end{aligned}

\end{gather}$

$\text{Therefore} \ e - e^{-1} \ \text{is our final answer.}$