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