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MAT334--2020F => MAT334--Tests and Quizzes => Quiz 4 => Topic started by: RunboZhang on October 23, 2020, 03:48:21 PM

Title: Quiz-5101-C
Post by: RunboZhang 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}

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