Proposition

Prove that $\int_{\gamma} z\exp(z^2) dz = 0$ for any closed path $\gamma$.

Solution

Let $f(z) = 1$ and $g(z) = \exp(z^2)/2$. Then $fg’ = z\exp(z^2)$.

By integration by parts, $\int_{\gamma} fg’ = f(\gamma(b))g(\gamma(b)) - f(\gamma(a))g(\gamma(a)) - \int_{\gamma}f’g = 0 - \int_{\gamma} 0 = 0$.