In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathematician Camille Jordan.
Consider a complex-valued, continuous function, defined on a semicircular contour
CR=\{Rei\mid\theta\in[0,\pi]\}
of positive radius lying in the upper half-plane, centered at the origin. If the function is of the form
f(z)=eig(z), z\inC,
with a positive parameter, then Jordan's lemma states the following upper bound for the contour integral:
\left|
\int | |
CR |
f(z)dz\right|\le
\pi | |
a |
MR where MR:=max\theta\left|g\left(Rei\right)\right|.
with equality when vanishes everywhere, in which case both sides are identically zero. An analogous statement for a semicircular contour in the lower half-plane holds when .