Author Topic: Quiz-5101-A  (Read 304 times)

Kuba Wernerowski

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Quiz-5101-A
« on: October 23, 2020, 11:17:53 AM »
Evaluate the given integral using the technique of Example 10 of Section 2.3: $$
\int_\gamma \frac{dz}{z^2},$$ where $\gamma$ is any curve in $\{ z: Re \, z \geq 0, z \neq 0 \},$ joining $-i$ to $1+i$.

Solution:

$F(z) = \frac{-1}{z}$, where $F'(z) = f(z) = \frac{1}{z^2}$.

Note that $F$ is analytic whenever $z \neq 0$. Therefore, $F$ is analytic on $\gamma$.

So we have $$\int_\gamma f(z) dz = \int_\gamma F'(z) dz$$
\begin{align*}
 \int_\gamma F'(z) dz &= F(\text{end point}) - F(\text{initial point}) \\
                                &= F(1 + i) - F(-i) \\
                                &= \frac{-1}{1+i} - \left(\frac{-1}{i}\right) \\
                                &= \frac{-i + 1 + i}{(1+i)i} \\
                                &= \frac{1}{i-1} \\
                                &= \frac{1}{i-1} \frac{i+1}{i+1} \\
                                &= -\frac{1 + i}{2} \\

\end{align*}
« Last Edit: October 23, 2020, 04:40:54 PM by Kuba Wernerowski »