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

Freitas, Pedro

doi:10.1090/mcom/3267

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

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