Problem: Evaluate the given integral using the technique of Example 10 of Section 2.3:
$$\int_\gamma \frac{dz}{z^3} , $$
where $\gamma$ is any curve in $\{z: Re z \geq 0, z \neq 0 \}$, joining $-i$ to $1 + i$.
Answer:
since f is analytic in all $Re(z) \geq 0, z \neq 0, \int_\gamma \frac{dz}{z^3} = F(1+i) - F(-i)$
$$ F'(z) = f(z) \Rightarrow F(z) = - \frac{1}{2z^2}$$
$$ F(1+i) = - \frac{1}{2(1+i)^2} = - \frac{1}{4i}$$
$$ F(-i) = - \frac{1}{2(-i)^2} = \frac{1}{2}$$
Thus, $ \int_\gamma \frac{dz}{z^3} = F(1+i) - F(-i) = - \frac{1}{4i} - \frac{1}{2} = \frac{2i-1}{4i}$
