Proposition

Evaluate $\int_{\gamma} \exp(3z) dz$ for each of the following paths:

$\gamma$ is the line segment from 1 to $i$
$\gamma = C[0, 3]$
$\gamma$ is the arc of the parabola $y = x^2$ from $x = 0$ to $x = 1$.

Solution

1

Let $\gamma(t) = 1 - t + it$ with $t \in [0, 1]$. By Theorem 4.11, it suffices to calculate $\frac{\exp(3\gamma(1)) - \exp(3\gamma(0))}{3} = (e^{3i} - e^3)/3$.

2

0 by Corollary 4.20.

3

Let $\gamma(t) = t + it^2$.

By Theorem 4.11, $\exp(1 + i) - 1) / 3$.