Jordan's Lemma
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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:
| \(\ds \size {\int_{C_r} \map f z \rd z}\) | \(=\) | \(\ds \size {\int_{C_r} e^{i a z} \map g z \rd z}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \size {i r} \size {\int_0^\pi e^{i a r e^{i \theta} } \map g {r e^{i \theta} } e^{i \theta} \rd \theta}\) | Definition of Complex Contour Integral | |||||||||||
| \(\ds \) | \(=\) | \(\ds \size {i r} \size {\int_0^\pi e^{i a r \paren {i \sin \theta + \cos \theta} } \map g {r e^{i \theta} } e^{i \theta} \rd \theta}\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds r \size {\int_0^\pi e^{a r \paren {i \cos \theta - \sin \theta} } \map g {r e^{i \theta} } e^{i \theta} \rd \theta}\) | $i^2 = -1$, $\size i = 1$ | |||||||||||
| \(\ds \) | \(\le\) | \(\ds r \int_0^\pi \size {e^{i a r \cos \theta} } \size {e^{-a r \sin \theta} } \size {\map g {r e^{i \theta} } } \size {e^{i \theta} } \rd \theta\) | Modulus of Complex Integral | |||||||||||
| \(\ds \) | \(=\) | \(\ds r \int_0^\pi e^{-a r \sin \theta} \size {\map g {r e^{i \theta} } } \rd \theta\) | $\size {e^{i \theta} } = 1$ for real $\theta$ | |||||||||||
| \(\ds \) | \(\le\) | \(\ds r \int_0^\pi e^{-a r \sin \theta} \max_{0 \le \theta \le \pi} \size {\map g {r e^{i \theta} } } \rd \theta\) | Definition of Supremum of Real-Valued Function |
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:
| \(\ds \int_0^\pi e^{-a r \sin \theta} \rd \theta\) | \(=\) | \(\ds \int_0^{\pi/2} e^{-a r \sin \theta} \rd \theta + \int_{\pi/2}^\pi e^{-a r \sin \theta} \rd \theta\) | Sum of Integrals on Adjacent Intervals for Integrable Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^{\pi/2} e^{-a r \sin \theta} \rd \theta - \int_{\pi/2}^0 e^{-a r \map \sin {\pi - \theta} } \rd \theta\) | substituting $\theta \mapsto \pi - \theta$]] | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^{\pi/2} e^{-a r \sin \theta} \rd \theta + \int_0^{\pi/2} e^{-a r \sin \theta} \rd \theta\) | Sine of Angle plus Integer Multiple of Pi, Sine Function is Odd | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \int_0^{\pi/2} e^{-a r \sin \theta} \rd \theta\) |
Note that we also have, from Jordan's Inequality:
- $\sin \theta \ge \dfrac {2 \theta} \pi$
We therefore have:
| \(\ds 2 \int_0^{\pi/2} e^{-a r \sin \theta} \rd \theta\) | \(\le\) | \(\ds 2 \int_0^\pi e^{-\paren {2 a r \theta} / \pi} \rd \theta\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \intlimits {-\frac {\pi e^{\paren {2 a r \theta} / \pi} } {2 a r} } 0 \pi\) | Primitive of Exponential of a x, Fundamental Theorem of Calculus | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\pi \paren {1 - e^{-2 a r} } } {a r}\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \frac \pi {a r}\) | $e^{-a r} < 1$ for positive $a, r$ |
So:
| \(\ds \size {\int_{C_r} \map f z \rd z}\) | \(\le\) | \(\ds r \paren {\max_{0 \le \theta \le \pi} \size {\map g {r e^{i \theta} } } } \int_0^\pi e^{-a r \sin \theta} \rd \theta\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \frac \pi {a r} \times r \times \paren {\max_{0 \le \theta \le \pi} \size {\map g {r e^{i \theta} } } }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac \pi a \paren {\max_{0 \le \theta \le \pi} \size {\map g {r e^{i \theta} } } }\) |
$\blacksquare$
Also see
Source of Name
This entry was named for Marie Ennemond Camille Jordan.