Table Of ContentTexts & Monographs in Symbolic Computation
Clemens G. Raab
Michael F. Singer Editors
Integration in
Finite Terms:
Fundamental
Sources
Texts & Monographs in Symbolic
Computation
A Series of the Research Institute for Symbolic
Computation, Johannes Kepler University, Linz,
Austria
Founding Editor
Bruno Buchberger, Linz, Austria
Series Editor
Peter Paule, Linz, Austria
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Clemens G. Raab · Michael F. Singer
Editors
Integration in Finite
Terms: Fundamental
Sources
Editors
Clemens G. Raab Michael F. Singer
Institute for Algebra Department of Mathematics
Johannes Kepler University of Linz North Carolina State University
Linz, Austria Raleigh, NC, USA
ISSN 0943-853X ISSN 2197-8409 (electronic)
Texts & Monographs in Symbolic Computation
ISBN 978-3-030-98766-4 ISBN 978-3-030-98767-1 (eBook)
https://doi.org/10.1007/978-3-030-98767-1
© Springer Nature Switzerland AG 2022
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Preface
TheFundamentalTheoremofCalculustellsusthatif 𝑓(𝑥) isaRiemannintegrable
functiononaninterval [𝑎,𝑏] and𝐹(𝑥)isafunctionwith d𝐹(𝑥) = 𝑓(𝑥),thatis,𝐹(𝑥)
d𝑥
isanantiderivativeof 𝑓(𝑥),then
∫
𝑏
𝑓(𝑥)d𝑥 = 𝐹(𝑏)−𝐹(𝑎).
𝑎
This theorem motivated the development and teaching of various heuristic tech-
niques for finding antiderivatives in order to evaluate definite integrals. In ba-
sic calculus courses, the foc√us is usually on the so-called elementary functions
(e𝑥,sin(𝑥),log(𝑥),arctan(𝑥), 𝑥,...)andtheseheuristictechniquesyieldantideriva-
tivesexpressibleagainintermsoftheseelementaryfunctions.Twonaturalquestions
arise:
1. Doallelementaryfunctionshaveanelementaryantiderivativeand,ifnot,how
doesonecharacterizethosethatdo?
2. Isthereanalgorithmtodetermineifanelementaryfunctionhasanelementary
antiderivativeand,ifso,canwefindone?
Liouville’sTheoremanswersthefirstquestionandcanbeusedtoshowthate𝑥2 does
nothaveanelementaryantiderivativeandRisch’sAlgorithmgivesapositiveanswer
tothesecondquestion.Theseresultsarenowcentraltoawell-developedareadevoted
to finding antiderivatives and, more generally, solutions of differential equations,
expressibleintermsofclassesofelementaryandspecialfunctions;thesubjectof
IntegrationinFiniteTerms.
Thisvolumepresentsfourofthefoundationaltextsrelatedtothistopicaswell
as commentaries discussing these texts and their impact. The first is a reprint of
MaxwellRosenlicht’spaperIntegrationinFiniteTerms[Ros72],presenting,inavery
elementaryway,apurelyalgebraicproofofLiouville’sTheorem.Acommentaryby
MichaelF.SingergivesanoutlineofthehistoryofLiouville’sTheoremanditsmodern
developments.ThesecondtextisareprintofJosephFelsRitt’sbookIntegrationin
FiniteTerms:Liouville’sTheoryofElementaryFunctions[Rit48].This1948book
v
Preface
presentsLiouville’s(andothers’)analyticapproachtothissubject.Itisfollowedby
acommentarybyAskholdKhovanskiipresentinghistopologicalapproachtothis
problem. The third is a revised version of Robert H. Risch’s unpublished On the
integrationofelementaryfunctionswhicharebuiltupusingalgebraicoperations
[Ris68]incorporatingunpublishedresultsof[Ris69].Thisreducedtheproblemof
findingelementaryantiderivativesofelementaryfunctionstoaprobleminarithmetic
algebraicgeometry:theproblemofdeterminingifapointonthejacobianofacurve
isoffiniteorder.Rischpresentedasolutionofthelatterproblemin[Ris70]inthe
contextofdecidingtheelementaryintegrabilityofanalgebraicfunction.ClemensG.
Raab’scommentarydiscussestheimpactofthispaperandfurtherdevelopments.The
finaltextisBarryM.Trager’sunpublishedthesisIntegrationofAlgebraicFunctions
[Tra84],whichprovidedpracticalalgorithmsforperformingtheintegrationinfinite
termsofalgebraicfunctionsafterthetheoreticalresultsofRisch.Itisfollowedbya
commentarybyBarryTrager,whogivesfurtherinsightintothemethodsofhisthesis
aswellassubsequentdevelopments.
SpecialthanksgotoPeterPaule,whoseencouragementandsupportwerecrucial
tobeginningthisprojectandseeingittocompletion.
References
[Ris68] RobertH.Risch,Ontheintegrationofelementaryfunctionswhicharebuilt
upusingalgebraicoperations,SDCDocumentSP-2801/001/00,June1968.
[Ris69] ,FurtherResultsonElementaryFunctions,IBMreportRC-2401,
1969.
[Ris70] , The solution of theproblem of integration in finite terms, Bull.
Amer.Math.Soc.76(1970),605–608.MR0269635
[Rit48] JosephF.Ritt,IntegrationinFiniteTerms.Liouville’sTheoryofElementary
Methods,ColumbiaUniversityPress,NewYork,N.Y.,1948.MR0024949
[Ros72] MaxwellRosenlicht,Integrationinfiniteterms,Amer.Math.Monthly79
(1972),963–972.MR0321914
[Tra84] Barry M. Trager, Integration of Algebraic Functions, Ph.D. thesis, Mas-
sachusettsInstituteofTechnology,1984.
vi
Contents
Preface ....................................................... v
IntegrationinFiniteTerms....................................... 1
MaxwellRosenlicht
CommentsonRosenlicht’sIntegrationinFiniteTerms ................ 11
MichaelF.Singer
IntegrationinFiniteTerms:Liouville’sTheoryofElementaryMethods . 31
JosephFelsRitt
CommentsonJ.F.Ritt’sBookIntegrationinFiniteTerms ............. 137
AskoldKhovanskii
OntheIntegrationofElementaryFunctionswhichareBuiltUpUsing
AlgebraicOperations ........................................... 203
RobertH.Risch
CommentsonRisch’sOntheIntegrationofElementaryFunctionswhich
areBuiltUpUsingAlgebraicOperations ............................ 221
ClemensG.Raab
IntegrationofAlgebraicFunctions ................................ 235
BarryM.Trager
CommentsonIntegrationofAlgebraicFunctions..................... 297
BarryM.Trager
vii
Integration in Finite Terms
MaxwellRosenlicht
Amer.Math.Monthly79(9),963–972(1972)
© Springer Nature Switzerland AG 2022
1
C. G. Raab und M. F. Singer (Hrsg.), Integration in Finite Terms:
Fundamental Sources, Texts & Monographs in Symbolic Computation,
https://doi.org/10.1007/978-3-030-98767-1_1
2
