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Sharp bounds for the modulus and phase of Hankel functions with applications to Jaeger integrals


Author: Pedro Freitas
Journal: Math. Comp.
MSC (2010): Primary 33C10, 26D15; Secondary 33E20, 35C15
DOI: https://doi.org/10.1090/mcom/3267
Published electronically: May 31, 2017
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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.


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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

DOI: https://doi.org/10.1090/mcom/3267
Keywords: Jaeger integrals, Hankel functions, Ramanujan integral, asymptotic expansions, sharp bounds
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
Article copyright: © Copyright 2017 American Mathematical Society