# oscillatory integrals

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##### 1: 36.12 Uniform Approximation of Integrals
The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. …
36.12.3 $I(\mathbf{y},k)=\frac{\exp\left(ikA(\mathbf{y})\right)}{k^{1/(K+2)}}\sum% \limits_{m=0}^{K}\frac{a_{m}(\mathbf{y})}{k^{m/(K+2)}}\left(\delta_{m,0}-\left% (1-\delta_{m,0}\right)i\frac{\partial}{\partial z_{m}}\right)\Psi_{K}\left(% \mathbf{z}(\mathbf{y};k)\right)\left(1+O\left(\frac{1}{k}\right)\right),$
36.12.11 $I(y,k)=\frac{\Delta^{1/4}\pi\sqrt{2}}{k^{1/3}}\exp\left(ik\widetilde{f}\right)% \left(\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime\prime}}}+\frac{g_{-}}{\sqrt{-f_{-% }^{\prime\prime}}}\right)\mathrm{Ai}\left(-k^{2/3}\Delta\right)\left(1+O\left(% \frac{1}{k}\right)\right)-i\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime\prime}}}-% \frac{g_{-}}{\sqrt{-f_{-}^{\prime\prime}}}\right)\frac{\mathrm{Ai}'\left(-k^{2% /3}\Delta\right)}{k^{1/3}\Delta^{1/2}}\left(1+O\left(\frac{1}{k}\right)\right)% \right),$
###### §3.5(vii) OscillatoryIntegrals
Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature. … …
##### 3: Bibliography V
• A. N. Varčenko (1976) Newton polyhedra and estimates of oscillatory integrals. Funkcional. Anal. i Priložen. 10 (3), pp. 13–38 (Russian).
• ##### 4: Bibliography E
• G. A. Evans and J. R. Webster (1999) A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.
• ##### 5: Bibliography W
• R. Wong (1982) Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.
• ##### 6: Bibliography D
• J. J. Duistermaat (1974) Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.
• ##### 7: Bibliography C
• J. N. L. Connor and P. R. Curtis (1982) A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: Application to Pearcey’s integral and its derivatives. J. Phys. A 15 (4), pp. 1179–1190.
##### 9: 2.3 Integrals of a Real Variable
For extensions to oscillatory integrals with logarithmic singularities see Wong and Lin (1978). …
##### 10: Bibliography L
• I. M. Longman (1956) Note on a method for computing infinite integrals of oscillatory functions. Proc. Cambridge Philos. Soc. 52 (4), pp. 764–768.