Proposition

Compute $\int_{C[0, 2]}\frac{\exp(z)}{(z - w)^2}$ where $w$ is any fixed complex number $\abs{w} \ne 2$.

Solution

If $\abs{w} < 2$, then by Cauchy’s Integral formula, the integral equals $2\pi i \exp(w)$. If $\abs{w} > 2$, then by Corollary 4.20[A first course in complex analysis, the integral equals 0.