Analytic formulas for the evaluation of the Pearcey integral

López, José; Pagola, Pedro

doi:10.1090/mcom/3164

Analytic formulas for the evaluation of the Pearcey integral
HTML articles powered by AMS MathViewer

by José L. López and Pedro J. Pagola PDF

Math. Comp. 86 (2017), 2399-2407 Request permission

Abstract:

We can find in the literature several convergent and/or asymptotic expansions of the Pearcey integral $P(x,y)$ in different regions of the complex variables $x$ and $y$, but they do not cover the whole complex $x$ and $y$ planes. The purpose of this paper is to complete this analysis giving new convergent and/or asymptotic expansions that, together with the known ones, cover the evaluation of the Pearcey integral in a large region of the complex $x$ and $y$ planes. The accuracy of the approximations derived in this paper is illustrated with some numerical experiments. Moreover, the expansions derived here are simpler compared with other known expansions, as they are derived from a simple manipulation of the integral definition of $P(x,y)$.

References

M. V. Berry and C. J. Howls, Integrals with coalescing saddles, NIST handbook of mathematical functions, U.S. Dept. Commerce, Washington, DC, 2010, pp. 775–793. MR 2655376
David Kaminski, Asymptotic expansion of the Pearcey integral near the caustic, SIAM J. Math. Anal. 20 (1989), no. 4, 987–1005. MR 1000733, DOI 10.1137/0520066
T. H. Koornwinder, R. Wong, R. Koekoek, and R. F. Swarttouw, Orthogonal polynomials, NIST handbook of mathematical functions, U.S. Dept. Commerce, Washington, DC, 2010, pp. 435–484. MR 2655358
José L. López and Nico M. Temme, Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions, Stud. Appl. Math. 103 (1999), no. 3, 241–258. MR 1712633, DOI 10.1111/1467-9590.00126
José L. López and Pedro J. Pagola, Convergent and asymptotic expansions of the Pearcey integral, J. Math. Anal. Appl. 430 (2015), no. 1, 181–192. MR 3347208, DOI 10.1016/j.jmaa.2015.04.078
José L. López and Pedro J. Pagola, The Pearcey integral in the highly oscillatory region, Appl. Math. Comput. 275 (2016), 404–410. MR 3437720, DOI 10.1016/j.amc.2015.11.080
Lord Kelvin, Deep water ship-waves, Philos. Mag. 9 (1905), 733–757.
A. B. Olde Daalhuis, On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles, Methods Appl. Anal. 7 (2000), no. 4, 727–745. MR 1868554, DOI 10.4310/MAA.2000.v7.n4.a7
A. B. Olde Daalhuis, Confluent hypergeometric functions, NIST handbook of mathematical functions, U.S. Dept. Commerce, Washington, DC, 2010, pp. 321–349. MR 2655353
R. B. Paris, The asymptotic behaviour of Pearcey’s integral for complex variables, Proc. Roy. Soc. London Ser. A 432 (1991), no. 1886, 391–426. MR 1116539, DOI 10.1098/rspa.1991.0023
R. B. Paris and D. Kaminski, Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions, J. Comput. Appl. Math. 190 (2006), no. 1-2, 437–452. MR 2209518, DOI 10.1016/j.cam.2005.01.038
T. Pearcey, The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic, Philos. Mag. (7) 37 (1946), 311–317. MR 20857
F. Ursell, Integrals with a large parameter. Several nearly coincident saddle-points, Proc. Cambridge Philos. Soc. 72 (1972), 49–65. MR 299995, DOI 10.1017/s0305004100050945
R. Wong, Asymptotic approximations of integrals, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. MR 1016818