Jordan's lemma

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Jordan's lemma in complex analysis is a powerful tool used frequently when evaluating contour integrals.

Consider an integral,

$I=\int_\Gamma e^{iaz} g(z) dz\,$

where $Γ$ is a semicircular path of radius $R$ centered at the origin of the complex plane and extending across the upper half plane, not including the diameter.

Jordan's lemma states that, $I \to 0$ as $R\rightarrow \infty,$ provided that $g(z) \rightarrow 0$ as $z \rightarrow \infty$ in the upper half plane and $a$ is a positive real number. If $a$ is negative, then the lower half plane must be used.

A typical application of Jordan's lemma is one where we want to evaluate an integral, which can be written in the form as shown below, along the real axis. In this case, the integral can be easily evaluated (using the residue theorem) by summing the residues in the upper or lower half planes depending on the sign of a.

$I=\int_{-\infty}^\infty e^{iaz} g(z) dz\,=2\pi i \sum r_{g(z)}$