Jordan's lemma

Statement

Let $a > 0$ and $R > 0$ be real numbers. Let $C_R$ be the semicircle $\{ R e^{i\theta} : \theta \in [0, \pi] \}$ in the upper half-plane, with the parametrization $\theta \mapsto R e^{i\theta}$. Let $g$ be a continuous complex function on $C_R$, and let $f(z) = e^{iaz} g(z)$ for $z \in C_R$, where $e^w$ is the complex exponential. Let $M_R = \max_{\theta \in [0, \pi]} |g(R e^{i\theta})|$. Then the contour integral of $f$ along $C_R$ satisfies $$\left| \int_{C_R} f(z) \, dz \right| \le \frac{\pi}{a} M_R.$$

Remarks

If $M_R \to 0$ as $R \to \infty$, then $\int_{C_R} f(z) \, dz \to 0$ as $R \to \infty$. The bound does not contain the length $\pi R$ of $C_R$, so it is stronger than the estimation lemma for this type of integrand. A similar result holds for the semicircle in the lower half-plane when $a < 0$. The lemma is a main step in the evaluation of real integrals such as $\int_{-\infty}^{\infty} \frac{e^{iax}}{x^2 + 1} \, dx$.

Explanations

L1 human

On the semicircle, $|e^{iaz}| = e^{-aR \sin \theta}$, and this number is very small except near the two ends of the semicircle. So the factor $e^{iaz}$ kills most of the long path. This is why the bound stays small even when the radius $R$ is large.