Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

The Asymptotic Behaviour of Pearcey's Integral for Complex Variables

R. B. Paris
Proceedings: Mathematical and Physical Sciences
Vol. 432, No. 1886 (Mar. 8, 1991), pp. 391-426
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/51818
Page Count: 36
  • Get Access
  • Read Online (Free)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
The Asymptotic Behaviour of Pearcey's Integral for Complex Variables
Preview not available

Abstract

A deeper understanding of the rich structure of the canonical form of the oscillatory integral describing the cusp diffraction catastrophe, generally known as Pearcey's integral P(X, Y), can be obtained by considering its analytic continuation to arbitrary complex variables X and Y. A new integral representation for P(X, Y) is given in the form of a contour integral involving a Weber parabolic cylinder function whose order is the variable of integration. It is shown how the asymptotics of P(X, Y) may be obtained from this representation for complex X and Y when either |X| or |Y|→ ∞ , without reference to the usual stationary points of the integrand. For the case |X|→ ∞ , Y finite the full asymptotic expansion of P(X, Y) is derived and its asymptotic character is found to be either exponentially large or algebraic in certain sectors of the X-plane. The case |Y|→ ∞ , X finite is complicated by the presence of exponentially small subdominant terms in certain sectors of the Y-plane, and only the first terms in the expansion are given. The asymptotic behaviour of P(X, Y) on the caustic Y2 + (2/3X)3 = 0 is also obtained from the new representation and is shown to agree with recent results of D. Kaminski. The various properties of the Weber parabolic cylinder function required in this paper are collected together in the Appendix.

Page Thumbnails

  • Thumbnail: Page 
391
    391
  • Thumbnail: Page 
392
    392
  • Thumbnail: Page 
393
    393
  • Thumbnail: Page 
394
    394
  • Thumbnail: Page 
395
    395
  • Thumbnail: Page 
396
    396
  • Thumbnail: Page 
397
    397
  • Thumbnail: Page 
398
    398
  • Thumbnail: Page 
399
    399
  • Thumbnail: Page 
400
    400
  • Thumbnail: Page 
401
    401
  • Thumbnail: Page 
402
    402
  • Thumbnail: Page 
403
    403
  • Thumbnail: Page 
404
    404
  • Thumbnail: Page 
405
    405
  • Thumbnail: Page 
406
    406
  • Thumbnail: Page 
407
    407
  • Thumbnail: Page 
408
    408
  • Thumbnail: Page 
409
    409
  • Thumbnail: Page 
410
    410
  • Thumbnail: Page 
411
    411
  • Thumbnail: Page 
412
    412
  • Thumbnail: Page 
413
    413
  • Thumbnail: Page 
414
    414
  • Thumbnail: Page 
415
    415
  • Thumbnail: Page 
416
    416
  • Thumbnail: Page 
417
    417
  • Thumbnail: Page 
418
    418
  • Thumbnail: Page 
419
    419
  • Thumbnail: Page 
420
    420
  • Thumbnail: Page 
421
    421
  • Thumbnail: Page 
422
    422
  • Thumbnail: Page 
423
    423
  • Thumbnail: Page 
424
    424
  • Thumbnail: Page 
425
    425
  • Thumbnail: Page 
426
    426