Jordan's Lemma

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Theorem

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

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

If the function $f$ is of the form:

$f \left({z}\right) = e^{i a z} g \left({z}\right), \ a > 0, \ z \in C_r$

Then:

$\displaystyle \left\vert{\int_{C_r} f \left({z}\right) \rd z}\right\vert \le \frac \pi a \max_{0 \le \theta \le \pi} \left\vert{g \left(re^{i \theta}\right)}\right\vert$


Proof

\(\displaystyle \left\vert{\int_{C_r} f\left({z}\right) \rd z}\right\vert\) \(=\) \(\displaystyle \left\vert{\int_{C_r} e^{i a z} g\left({z}\right) \rd z}\right\vert\)
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{i r}\right\vert \left\vert{\int_0^\pi e^{i a r e^{i \theta} } g \left({r e^{i \theta} }\right) e^{i \theta} \rd \theta}\right\vert\) Definition of Complex Contour Integral
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{i r}\right\vert \left\vert{\int_0^\pi e^{i a r \left({i \sin \theta + \cos \theta}\right)} g \left({r e^{i \theta} }\right) e^{i \theta} \rd \theta}\right\vert\) Euler's Formula
\(\displaystyle \) \(=\) \(\displaystyle r \left\vert{\int_0^\pi e^{a r \left({i \cos \theta - \sin \theta}\right)} g \left({r e^{i \theta} }\right) e^{i \theta} \rd \theta}\right\vert\) $i^2 = -1$, $\left\vert{i}\right\vert = 1$
\(\displaystyle \) \(\le\) \(\displaystyle r \int_0^\pi \left\vert{e^{i a r \cos \theta} }\right\vert \left\vert{e^{-a r \sin \theta} }\right\vert \left\vert{g \left({r e^{i \theta} }\right)}\right\vert \left\vert{e^{i \theta} }\right\vert \rd \theta\) Modulus of Complex Integral
\(\displaystyle \) \(=\) \(\displaystyle r \int_0^\pi e^{-a r \sin \theta} \left\vert{g \left({r e^{i\theta} }\right)}\right\vert \rd \theta\) $\left\vert{e^{i \theta} }\right\vert = 1$ for real $\theta$
\(\displaystyle \) \(\le\) \(\displaystyle r \int_0^\pi e^{-a r \sin \theta} \max_{0 \le \theta \le \pi} \left\vert{g \left({r e^{i \theta} }\right)}\right\vert \rd \theta\) Definition of Supremum of Real-Valued Function
\(\displaystyle \) \(\le\) \(\displaystyle r \max_{0 \le \theta \le \pi} \left\vert{g \left({r e^{i \theta} }\right)}\right\vert \int_0^\pi e^{-\left({2 a r \theta}\right) / \pi} \rd \theta\) Shape of Sine Function
\(\displaystyle \) \(=\) \(\displaystyle r \max_{0 \le \theta \le \pi} \left\vert{g\left({r e^{i \theta} }\right)}\right\vert \left[{- \frac{\pi e^{\left({2 a r \theta}\right) / \pi} } {2 a r} }\right]_0^\pi\) Primitive of Exponential of a x, Fundamental Theorem of Calculus
\(\displaystyle \) \(=\) \(\displaystyle \max_{0 \le \theta \le \pi} \left\vert{g\left({r e^{i \theta} }\right)}\right\vert \frac{\pi \left({1 - e^{-2 a r} }\right)} {2 a}\)
\(\displaystyle \) \(\le\) \(\displaystyle \frac \pi {2 a} \max_{0 \le \theta \le \pi} \left\vert{g\left({r e^{i \theta} }\right)}\right\vert\) $e^{-ar} < 1$ for positive $a, r$
\(\displaystyle \) \(\le\) \(\displaystyle \frac \pi a \max_{0 \le \theta \le \pi} \left\vert{g\left( {r e^{i \theta} }\right)}\right\vert\)

$\blacksquare$


Also see


Source of Name

This entry was named for Marie Ennemond Camille Jordan.