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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. …
36.12.1 I ( y , k ) = - exp ( i k f ( u ; y ) ) g ( u , y ) d u ,
36.12.3 I ( y , k ) = exp ( i k A ( y ) ) k 1 / ( K + 2 ) m = 0 K a m ( y ) k m / ( K + 2 ) ( δ m , 0 - ( 1 - δ m , 0 ) i z m ) Ψ K ( z ( y ; k ) ) ( 1 + O ( 1 k ) ) ,
36.12.11 I ( y , k ) = Δ 1 / 4 π 2 k 1 / 3 exp ( i k f ~ ) ( ( g + f + ′′ + g - - f - ′′ ) Ai ( - k 2 / 3 Δ ) ( 1 + O ( 1 k ) ) - i ( g + f + ′′ - g - - f - ′′ ) Ai ( - k 2 / 3 Δ ) k 1 / 3 Δ 1 / 2 ( 1 + O ( 1 k ) ) ) ,
2: 3.5 Quadrature
§3.5(vii) Oscillatory Integrals
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.
  • 8: 36.2 Catastrophes and Canonical Integrals
    §36.2 Catastrophes and Canonical Integrals
    §36.2(i) Definitions
    Canonical Integrals
    §36.2(iii) Symmetries
    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.