Jordan's Lemma

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


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

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:


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

Also see

 * Estimation Lemma