# Jordan's Lemma

## 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 $a, r$ $\displaystyle$ $\le$ $\displaystyle \frac \pi a \max_{0 \le \theta \le \pi} \left\vert{g\left( {r e^{i \theta} }\right)}\right\vert$

$\blacksquare$

## Source of Name

This entry was named for Marie Ennemond Camille Jordan.