Jordan's Lemma

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


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

If the function $f$ is of the form:


 * $\map f z = e^{i a z} \map g z, \ a > 0, \ z \in C_r$

Then:


 * $\ds \size {\int_{C_r} \map f z \rd z} \le \frac \pi a \max_{0 \mathop \le \theta \mathop \le \pi} \size {\map g {r e^{i \theta} } }$

Also see

 * Estimation Lemma