Error bounds for uniform asymptotic expansions—modified bessel function of purely imaginary order

Abstract

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals. By using a class of rational functions, they express these quantities in terms of Cauchy-type integrals; these expressions are natural generalizations of integral representations of the coefficients and the remainders in the Taylor expansions of analytic functions. By using the new representation, a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.

Keywords

Modified Bessel function of purely imaginary order / Airy function / Uniform asymptotic expansion, Error bound

References

[1] Balogh C. B.. Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math., 1967, 15: 1315-1323

[2] Berry M. V.. Tsunami asymptotics. New J. Phys, 2005, 7: 1-18

[3] Bleistein N.. Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math., 1966, 19: 353-370

[4] Chester C., Friedman B., Ursell F.. An extension of the method of steepest descents. Proc. Cambridge Philos. Soc., 1957, 53: 599-611

[5] Friedlander F. G.. Diffraction of pulses by a circular cylinder. Comm. Pure Appl. Math., 1954, 7: 705-732

[6] Lear J. D., Sturm J. E.. An integral representation for the modified Bessel function of the third kind, computable for large, imaginary order. Math. Comput., 1967, 21(496): 496-498

[7] Lebedev N. N.. Special Function and Their Applications, 1965, Englewood Cliffs, New Jersey: Prentice-Hall

[8] Monroe C. W., Daikhin L. I., Urbakh M. Principles of electrowetting with two immiscible electrolytic solutions. J. Phys.: Condens. Matter, 2006, 18: 2837-2869

[9] Olver F. W. J.. Asymptotics and Special Functions, 1974, New York: Academic Press

[10] Olde Daalhuis A. B., Temme N. M.. Uniform Airy-type expansions of integrals. SIAM J. Math. Anal., 1994, 25: 304-321

[11] Oughstun K. E., Sherman G. C.. Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium). J. Opt. Soc. Amer. B, 1988, 5(4): 817-849

[12] Vidunas R., Temme N. M.. Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl., 2002, 269: 317-331

[13] Ådnøy Ellingsen S., Brevik I.. Electrodynamic Casimir effect in a medium-filled wedge. II. Physical Review, 2009, E80: 1-12

[14] Ursell F.. On Kelvins ship-wave pattern. J. Fluid Mech., 1960, 8: 418-431

[15] Wong R.. Error bound for asymptotic expansions of integrals. SIAM Review, 1980, 22: 401-435

[16] Wong R.. Asymptotic Approximations of Integrals, 1989, Boston: Academic Press