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Handtohand combat with thousanddigit integrals
, 2010
"... In this paper we describe numerical investigations of definite integrals that arise by considering the moments of multistep uniform random walks in the plane, together with a closely related class of integrals involving the elliptic functions K, K ′ , E and E ′. We find that in many cases such inte ..."
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Cited by 6 (3 self)
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In this paper we describe numerical investigations of definite integrals that arise by considering the moments of multistep uniform random walks in the plane, together with a closely related class of integrals involving the elliptic functions K, K ′ , E and E ′. We find that in many cases such integrals can be “experimentally ” evaluated in closed form or that intriguing linear relations exist within a class of similar integrals. Discovering these identities and relations often requires the evaluation of integrals to extreme precision, combined with largescale runs of the “PSLQ ” integer relation algorithm. This paper presents details of the techniques used in these calculations and mentions some of the many difficulties that can arise.
Crandall’s computation of the incomplete Gamma Function and the Hurwitz Zeta Function with applications to Dirichlet Lseries
, 2014
"... This paper extends tools developed by Richard Crandall in [16] to provide robust, highprecision methods for computation of the incomplete Gamma function and the Lerch transcendent. We then apply these to the corresponding computation of the Hurwitz zeta function and so of Dirichlet Lseries and cha ..."
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Cited by 2 (0 self)
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This paper extends tools developed by Richard Crandall in [16] to provide robust, highprecision methods for computation of the incomplete Gamma function and the Lerch transcendent. We then apply these to the corresponding computation of the Hurwitz zeta function and so of Dirichlet Lseries and character polylogarithms.
doi:10.1155/2007/19381 Research Article
"... New simple nestedsum representations for powers of the arcsin function are given. This generalization of Ramanujan’s workmakes connections to finite binomial sums and polylogarithms. Copyright © 2007 J. M. Borwein and M. Chamberland. This is an open access article distributed under the Creative C ..."
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New simple nestedsum representations for powers of the arcsin function are given. This generalization of Ramanujan’s workmakes connections to finite binomial sums and polylogarithms. Copyright © 2007 J. M. Borwein and M. Chamberland. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
Remarks on Slater’s Asymptotic Expansions of Kummer Functions for Large Values of the aParameter
"... To the memory of Panayiotis D. Siafarikas. The man who loved special functions. In Slater’s 1960 standard work on confluent hypergeometric functions, also called Kummer functions, a number of asymptotic expansions of these functions can be found. We summarize expansions derived from a differential e ..."
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To the memory of Panayiotis D. Siafarikas. The man who loved special functions. In Slater’s 1960 standard work on confluent hypergeometric functions, also called Kummer functions, a number of asymptotic expansions of these functions can be found. We summarize expansions derived from a differential equation for large values of the aparameter. We show how similar expansions can be derived by using integral representations, and we observe discrepancies with Slater’s expansions.
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, 2012
"... Abstract: We describe a general computational scheme for evaluation of a wide class of numbertheoretical functions. We avoid asymptotic expansions in favor of manifestly convergent series that lend themselves naturally to rigorous error bounds. By employing three fundamental series algorithms we ac ..."
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Abstract: We describe a general computational scheme for evaluation of a wide class of numbertheoretical functions. We avoid asymptotic expansions in favor of manifestly convergent series that lend themselves naturally to rigorous error bounds. By employing three fundamental series algorithms we achieve a unified strategy to compute the various functions via parameter selection. This work amounts to a compendium of methods to establish extremeprecision results as typify modern experimental mathematics. A fortuitous byproduct of this unified approach is automatic analytic continuation over complex parameters. Another byproduct is a host of converging series for various fundamental constants.