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:


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

Then:


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

Also see

 * Estimation Lemma