In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals.

Quotes

• Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject. Besides the obvious fact that the Fourier transform is itself an oscillatory integral par excellence, one need only recall the occurrence of Bessel functions in the work of Fourier, the study of asymptotics related to these functions by Airy, Stokes, and Lipschitz, and Riemann's use of the method of “stationary phase” in finding the asymptotics of certain Fourier transforms, all of which took place well over 100 years ago.
  • Elias M. Stein; Timothy S. Murphy (1993). Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press. p. 329. ISBN 0-691-03216-5. 

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