Jordan's Lemma

Theorem
Let $r > 0$ be a real number.

Let:


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

Let $f : C_r \to \C$ be a continuous function such that:


 * $\map f z = e^{i a z} \map g z$

for each $z \in C_r$, for some continuous function $g : C_r \to \C$ and real number $a > 0$.

Then:


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

Also see

 * Estimation Lemma