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 $g : C_r \to \C$ be a continuous function.

Define $f : C_r \to \C$ by:


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

for each $z \in C_r$, for some real number $a > 0$.

Then:


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

Proof
We have:

From Definite Integral of Constant Multiple of Real Function, we can write:


 * $\ds r \int_0^\pi e^{-a r \sin \theta} \max_{0 \le \theta \le \pi} \size {\map g {r e^{i \theta} } } \rd \theta = r \paren {\max_{0 \le \theta \le \pi} \size {\map g {r e^{i \theta} } } } \int_0^\pi e^{-a r \sin \theta} \rd \theta$

We now focus on:


 * $\ds \int_0^\pi e^{-a r \sin \theta} \rd \theta$

We have:

Note that we also have, from Jordan's Inequality:


 * $\sin \theta \ge \dfrac {2 \theta} \pi$

We therefore have:

So:

Also see

 * Estimation Lemma for Contour Integrals