Toronto Math Forum
MAT3342018F => MAT334Tests => Quiz4 => Topic started by: Victor Ivrii on October 26, 2018, 05:57:09 PM

$\renewcommand{\Re}{\operatorname{Re}}$
$\renewcommand{\Im}{\operatorname{Im}}$
Evaluate the given integral using the technique of Example 10 of Section 2.3 of the Textbook;
indicate which theorem or result you used to obtain your answer.
$$
\int_\gamma \frac{dz}{z^2},
$$
where $\gamma$ is any curve in $\{z\colon \Re z>0\}$ joining $(1i)$ to $(1+i)$.

\begin{equation*}
F( z) \ =\ \frac{1}{z}\\
\\
The\ function\ is\ analytic\ in\ all\ of\ Re\ z\ >\ 0\\
\\
So\ we\ just\ want\ F( end\ point) \ \ F( first\ point)\\
\\
F( 1i) \ \ F( 1\ +\ i)\\
\\
=\ \frac{1}{1i} \ +\ \frac{1}{1+i}\\
\\
=\ \frac{1\ \ i\ +\ 1\ \ i\ }{( 1i)( 1+i)} \ \\
\\
=\ \frac{2i}{2}\\
\\
=\ i
\end{equation*}