Jordan's lemma

In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. It is named after the French mathematician Camille Jordan.

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

of positive radius $R$ lying in the upper half-plane, centred at the origin. If the function $f$ is of the form

with a positive parameter $a$ , then Jordan's lemma states the following upper bound for the contour integral:

where equal sign is when $g$ vanishes everywhere. An analogous statement for a semicircular contour in the lower half-plane holds when $a < 0$ .

$\lim _{R\to \infty }M_{R}=0$

()

Jordan's lemma yields a simple way to calculate the integral along the real axis of functions $f (z) = e i a z g (z)$ holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points $z 1$ , $z 2$ , …, $z n$ . Consider the closed contour $C$ , which is the concatenation of the paths $C 1$ and $C 2$ shown in the picture. By definition,

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