Sharp bounds for the modulus and phase of Hankel functions with applications to Jaeger integrals
HTML articles powered by AMS MathViewer
- by Pedro Freitas PDF
- Math. Comp. 87 (2018), 289-308 Request permission
Abstract:
We prove upper and lower bounds for a class of Jaeger integrals $\mathcal {G}_{\nu }(\tau )$ appearing in axisymmetric diffusive transport related to several physical applications. In particular, we show that these integrals are globaly bounded either from above or from below by the first terms in their corresponding asymptotic expansions in $\tau$, both at zero and infinity. In the case of $\mathcal {G}_{0}(\tau )$ we show that it is bounded from below by the Ramanujan integral.
These bounds are obtained as a consequence of sharp bounds derived for the modulus and phase of Hankel functions, and for the Ramanujan integral, which we believe to be new and of independent interest, complementing the asymptotic and numerical results in the literature.
References
- R. A. Askey and R. Roy, Gamma function, in NIST Handbook of Mathematical Functions, edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, Cambridge University Press (2010).
- Árpád Baricz, Tibor K. Pogány, Saminathan Ponnusamy, and Imre Rudas, Bounds for Jaeger integrals, J. Math. Chem. 53 (2015), no. 5, 1257–1273. MR 3331903, DOI 10.1007/s10910-015-0485-7
- C. J. Bouwkamp, Note on an asymptotic expansion, Indiana Univ. Math. J. 21 (1971/72), 547–549. MR 291703, DOI 10.1512/iumj.1971.21.21043
- D. Britz, O. Østerby and J. Strutwolf, Reference values of the chronoamperometric response at cylindrical and capped cylindrical electrodes, Electrochimica Acta 55 (2010), 5629–5635.
- J. J. Dorning, B. Nicolaenko, and J. K. Thurber, An integral identity due to Ramanujan which occurs in neutron transport theory, J. Math. Mech. 19 (1969/1970), 429–438. MR 0254298, DOI 10.1512/iumj.1970.19.19040
- G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1940. MR 0004860
- Philip Hartman, On differential equations and the function $J_{\mu }{}^{2}+Y_{\mu }{}^{2}$, Amer. J. Math. 83 (1961), 154–188. MR 123039, DOI 10.2307/2372726
- J. C. Jaeger, Heat flow in the region bounded internally by a circular cylinder, Proc. Roy. Soc. Edinburgh Sect. A 61 (1942), 223–228. MR 7473
- J. C. Jaeger and Martha Clarke, A short table of $\int ^\infty _0(e^{-xu^2}/(J_0^2(u)+Y^2_0(u)))(du/u)$, Proc. Roy. Soc. Edinburgh Sect. A 61 (1942), 229–230. MR 7492
- Stefan G. Llewellyn Smith, The asymptotic behaviour of Ramanujan’s integral and its application to two-dimensional diffusion-like equations, European J. Appl. Math. 11 (2000), no. 1, 13–28. MR 1750387, DOI 10.1017/S0956792599004039
- J. W. Nicholson, A problem in the theory of heat conduction, Proc. Roy. Soc. London A 100 (1921), 226–240.
- F. W. J. Olver and L.C. Maximon, Bessel functions, in NIST Handbook of Mathematical Functions, edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, Cambridge University Press (2010).
- W. R. C. Phillips and P. J. Mahon, On approximations to a class of Jaeger integrals, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2136, 3570–3589. MR 2853295, DOI 10.1098/rspa.2011.0301
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
- J. Ernest Wilkins Jr., Nicholson’s integral for $J_n^2(z)+Y_n^2(z)$, Bull. Amer. Math. Soc. 54 (1948), 232–234. MR 25022, DOI 10.1090/S0002-9904-1948-08987-X
Additional Information
- Pedro Freitas
- Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal — and — Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016 Lisboa, Portugal
- Email: psfreitas@fc.ul.pt
- Received by editor(s): May 10, 2016
- Received by editor(s) in revised form: September 18, 2016
- Published electronically: May 31, 2017
- Additional Notes: The author was partially supported by FCT (Portugal) through project PTDC/MAT-CAL/4334/2014
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 289-308
- MSC (2010): Primary 33C10, 26D15; Secondary 33E20, 35C15
- DOI: https://doi.org/10.1090/mcom/3267
- MathSciNet review: 3716197