Functions of a complex variable Singularities and residues Lemma

Jordan's lemma

Builds on Contour integral Complex exponential

Statement

Let $a > 0$ and $R > 0$ be real numbers. Let $C_R$ be the semicircle $\{ R e^{i\theta} : \theta \in [0, \pi] \}$ in the upper half-plane, with the parametrization $\theta \mapsto R e^{i\theta}$ . Let $g$ be a continuous complex function on $C_R$ , and let $f(z) = e^{iaz} g(z)$ for $z \in C_R$ , where $e^w$ is the complex exponential. Let $M_R = \max_{\theta \in [0, \pi]} |g(R e^{i\theta})|$ . Then the contour integral of $f$ along $C_R$ satisfies $\left| \int_{C_R} f(z) \, dz \right| \le \frac{\pi}{a} M_R.$

Remarks

If $M_R \to 0$ as $R \to \infty$ , then $\int_{C_R} f(z) \, dz \to 0$ as $R \to \infty$ . The bound does not contain the length $\pi R$ of $C_R$ , so it is stronger than the estimation lemma for this type of integrand. A similar result holds for the semicircle in the lower half-plane when $a < 0$ . The lemma is a main step in the evaluation of real integrals such as $\int_{-\infty}^{\infty} \frac{e^{iax}}{x^2 + 1} \, dx$ .

Reading path

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Complex numbers object
Complex exponential definition
Contour integral definition
Jordan's lemma statement

Tag	`0PNPG2PP9H`
URI	`m2://0PNPG2PP9H`
Aliases	`jordan-lemma`
Kind	statement, lemma
Writing	W1 stated
MSC	30E20
Status	This copy of the site has no build data. Run `m2 status 0PNPG2PP9H`.
Uses	Contour integral, Complex exponential
Same as	Wikidata Q1816932 Wikipedia Jordan's lemma

Jordan's lemma

Statement

Remarks

Reading path

Further reading

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