Paper Info
Title
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 Weierstrass substition, a better expression can be found that is continuous on wider intervals than the first expression and yet is still an antiderivative of the integrand. The origin of the discontinuities is identified, and an algorithm is presented for automatically finding the improved type of antiderivative. The new algorithm also enlarges the set of functions that can be used in the substitution. The algorithm works by first evaluating the given integral using the Weierstrass substitution in the usual way and then removing any spurious discontinuities present in the antiderivative.
Year
DOI
Venue
1994
10.1145/174603.174409
ACM Trans. Math. Softw.
Keywords
Field
DocType
better expression,spurious discontinuity,computer-algebra system,trigonometric substitution,new algorithm,symbolic integration,following sense,continuity,computer algebra,trigonometric integral,algorithm work,improved type,weierstrass substition,weierstrass substitution
Tangent half-angle substitution,Mathematical optimization,Symbolic integration,Classification of discontinuities,Mathematical analysis,Nonelementary integral,Trigonometric substitution,Antiderivative (complex analysis),Antiderivative,Mathematics,Trigonometric integral
Journal
Volume
Issue
ISSN
20
1
0098-3500
Citations 
PageRank 
References 
6
1.82
0
Authors
2
Name
Order
Citations
PageRank
D. J. Jeffrey1338.39
A. D. Rich2143.90