Jordan's Lemma

Theorem
Consider a complex-valued, continuous function $f$ defined on the contour,


 * $C_r = \left\{re^{i\theta}:0\le \theta \le \pi\right\}, \ r>0$

If the function $f$ is of the form,


 * $f\left(z\right) = e^{iaz}g\left(z\right), \ a > 0, \ z \in C_r$

Then,
 * $\displaystyle \left|\int_{C_r} f(z)\mathrm dz\right| \le \frac \pi a \max_{0\le\theta\le\pi} \left|g\left(re^{i\theta}\right)\right|$

Also see

 * Estimation Lemma