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Title: Experimental computation with oscillatory integrals

Abstract

A previous study by one of the present authors, together with D. Borwein and I. Leonard [8], studied the asymptotic behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x and along the way looked at closed forms for integer values of p. In this study we address these integrals with the tools of experimental mathematics, namely by computing their numerical values to high precision, both as a challenge in itself, and also in an attempt to recognize the numerical values as closed-form constants. With this approach, we are able to reproduce several of the results of [8] and to find new results, both numeric and analytic, that go beyond the previous study.

Authors:
;
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
Computational Research Division
OSTI Identifier:
964382
Report Number(s):
LBNL-2158E
TRN: US200920%%274
DOE Contract Number:
DE-AC02-05CH11231
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97; INTEGRALS; NUMERICAL SOLUTION; ANALYTICAL SOLUTION; ACCURACY

Citation Formats

Bailey, David H., and Borwein, Jonathan M.. Experimental computation with oscillatory integrals. United States: N. p., 2009. Web. doi:10.2172/964382.
Bailey, David H., & Borwein, Jonathan M.. Experimental computation with oscillatory integrals. United States. doi:10.2172/964382.
Bailey, David H., and Borwein, Jonathan M.. 2009. "Experimental computation with oscillatory integrals". United States. doi:10.2172/964382. https://www.osti.gov/servlets/purl/964382.
@article{osti_964382,
title = {Experimental computation with oscillatory integrals},
author = {Bailey, David H. and Borwein, Jonathan M.},
abstractNote = {A previous study by one of the present authors, together with D. Borwein and I. Leonard [8], studied the asymptotic behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x and along the way looked at closed forms for integer values of p. In this study we address these integrals with the tools of experimental mathematics, namely by computing their numerical values to high precision, both as a challenge in itself, and also in an attempt to recognize the numerical values as closed-form constants. With this approach, we are able to reproduce several of the results of [8] and to find new results, both numeric and analytic, that go beyond the previous study.},
doi = {10.2172/964382},
journal = {},
number = ,
volume = ,
place = {United States},
year = 2009,
month = 6
}

Technical Report:

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