1994
DOI: 10.1145/174603.174409
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The evaluation of trigonometric integrals avoiding spurious discontinuities

Abstract: The tan(x/2) substitution, also called the Weierstrass substitution, is one method currently used by computer-algebra systems for the evaluation of trigonometric integrals. The method needs to be improved, because the expressions obtained using it sometimes contain discontinuities, which unnecessarily limit the domains over which the expressions are correct. We show that the discontinuities are spurious in the following sense: Given an integrand and an expression for its antiderivative that was obtained by the… Show more

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Cited by 20 publications
(6 citation statements)
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“…whereà ¼ B cosð/Þ=2 À A=2,B ¼ ÀB=2 sinð/Þ, c 0 ¼ 12=35M 2 ð1 þÃÞ À2 À a À1 p À1 ð0Þ correspond to pressure normalization p ¼ 0 at n ¼ 0, bc is the floor function which is added to remove spurious discontinuities associated with the inverse tangent function, 28 …”
Section: à3mentioning
confidence: 99%
“…whereà ¼ B cosð/Þ=2 À A=2,B ¼ ÀB=2 sinð/Þ, c 0 ¼ 12=35M 2 ð1 þÃÞ À2 À a À1 p À1 ð0Þ correspond to pressure normalization p ¼ 0 at n ¼ 0, bc is the floor function which is added to remove spurious discontinuities associated with the inverse tangent function, 28 …”
Section: à3mentioning
confidence: 99%
“…Progress has also been made on integration, particularly with the papers of Jeffrey (1993Jeffrey ( , 1994Jeffrey ( , 1997 and Jeffrey and Rich (1994). The main contribution of these papers is that it is better to return an integral that is continuous on a domain of maximum extent, rather than trying to fix up spurious singularities and branch cuts later.…”
Section: Summary Acknowledgement and Note Added In March 2000mentioning
confidence: 99%
“…. stands for the floor function, which has been added to remove spurious discontinuities associated with the arctan function (Jeffrey & Rich 1994), and pressure is normalized with condition p−1 ( 0) = 0. To extract the mean pressure gradient, observe that dp −1 /dξ in (4.6a) is an even function of ξ .…”
Section: Discussionmentioning
confidence: 99%
“…The resulting sequence of problems begins with the leading-order system being of the form Its solution can be readily determined, i.e. Integration of (A7 a ) results in an expression for pressure in the form of where In the above, symbol stands for the floor function, which has been added to remove spurious discontinuities associated with the arctan function (Jeffrey & Rich 1994), and pressure is normalized with condition . To extract the mean pressure gradient, observe that in (4.6 a ) is an even function of .…”
Section: Figure 29mentioning
confidence: 99%