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*}
