Jordan's lemma

Jordan's lemma bounds integrals over large semicircular contours and shows they vanish when an exponential factor decays on the arc and the remaining function satisfies suitable conditions.

Contour integration: Integration of complex-valued functions along oriented paths in the complex plane. Jordan's lemma supplies an estimate for one part of a closed contour.
Residue theorem: A theorem that evaluates contour integrals of meromorphic functions by summing residues at enclosed poles. After the arc vanishes, residues determine the real-axis integral.
Complex exponential: The exponential function extended to complex arguments, with magnitude and phase determined by the real and imaginary parts. Its magnitude on a semicircle creates the angular decay used by the lemma.
ML inequality: A bound on a contour integral by the path length times the maximum modulus of its integrand. The basic bound often fails to show vanishing when the arc length grows.
Cauchy's integral theorem: A theorem stating that a holomorphic function has zero integral around a closed contour in a simply connected domain. It makes the completed contour integral depend on enclosed singularities rather than the arc's path.
Fourier transform: A transformation that represents a function in terms of its frequency components. Jordan's lemma supports contour methods for integrals containing Fourier oscillations.
Holomorphic function: A complex function that is complex differentiable throughout an open set. Contour methods require holomorphicity away from isolated singularities.
Residue calculus: The use of residues and contour integration to evaluate complex and real integrals. Jordan's lemma complements residue calculations by controlling the contour's distant boundary.
Semicircular contour: A contour consisting of a straight segment and a semicircular arc joining its endpoints. The lemma estimates the arc as its radius grows.
Fourier inversion theorem: A theorem recovering a function from its Fourier transform under suitable hypotheses. Contour evaluation of oscillatory integrals can establish inversion formulas in particular cases.

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