From: tycchow@riesz.mit.edu (Timothy Y. Chow)
Subject: Re: algo. determining closed form integral from c.f. function
Organization: None.  This saves me from writing a disclaimer.
Date: Fri, 14 Aug 92 01:58:20 GMT

In article <israel.713747592@unixg.ubc.ca> israel@unixg.ubc.ca (Robert B. Israel) writes:

>AUTHOR:     Davenport, James Harold, 1953-
>TITLE:      On the integration of algebraic functions / James Harold 
>             Davenport. --
>CALL NO:    QA 341 D33 1981
>PUBLISHED:  Berlin ; New York : Springer-Verlag, 1981.
>DESCRIPTION: 197 p. ; 25 cm. -- Bibliography: p. [186]-197.
>SERIES:     Lecture notes in computer science ; 102


Following are some other references that I collected when I asked this
question some time ago:


<Date: Wed, 29 Apr 92 11:20:48 -0500
<From: Bruce E Litow <litow@csd4.csd.uwm.edu>
<
<Indeed, not so elementary. Do the names Abel, Galois and Liouville mean
<anything? A good survey is in "Computer Algebra", by Davenport, Siret
<and Tournier, Academic Press, 1988
<
<---------------------------------------------------------------------------
<
<Date: Wed, 29 Apr 92 22:44:04 -0400
<From: tga@math.appstate.edu (Terry Anderson)
<
<        The book "DEs with Applications and Historical Notes" by George F.
<Simmons (2nd edition, McGraw Hill) contains a note on Joseph Liouville
<(1809-1882), who studied the question of integration in finite terms.
<
<        References given are:
<
<D. G. Mead, "Integration," American Mathematical Monthly, vol. 68,  pp.
<152-156 (1961).     (QA 1 .A515)
<
<G. H. Hardy,  The Integration of Functions of a Single Variable,
<Cambridge Univ ersity Press,   London, 1916.
<
<J. F. Ritt,  Integration in Finite Terms,  Columbia University Press,
<New York, 1948.          (QA  308   .R5)
<
<Simmons' book also has extensive historical notes on Newton, Gauss, and
<Euler as well as short notes on Fermat, the Bernoulli family, Riemann,
<Laplace, Abel, Poincare', and others. More recent references on the
<solution of DEs in cl osed form are given in
<
<"Computer Algebra:  Past and Future," by B. F. Caviness, Journal of
<Symbolic Computation (1986), 217-236 (see pp. 225-6, in particular).
<
<J. J. Kovacic, "An Algorithm for solving 2nd order linear homogeneous
<DEs," manuscript (1979).   Also in J. Symbolic Computation 2, (1986),
<3-43.
<
<M. F. Singer,  "Functions satisfying elementary relations,"
<Transactions of the AMS 227                        (1977), 185-206.
<
<M. F. Singer,  "Liouvillian solutions of nth order homogeneous linear
<DEs,"  Amer. J. Math. 103                 (1981), 661-682.
<
<M. F. Singer, B. D. Saunders, and B. F. Caviness,  "An extension of
<Liouville's theorem on integration in finite terms,"  SIAM J. Comput.
<14 (1985), 966-990.
<
<B. D. Saunders,  "An implementation of Kovacic's algorithm for solving
<second order linear homogeneous DEs,"   in Proc. 1981 ACM Symp. on
<Symbolic and Algebraic Computation (editor P. S. Wang), pp. 105-108.
<
<R. H. Risch - several papers on integration in finite terms.         
<
<M. Rosenlicht,  "On Liouville's theory of elementary functions,"
<Pacific J. Math. 65 (1976), 485-492.
<
<B. W. Char,  "Using Lie transformation groups to find closed form
<solutions to first order ordinary differential equations,"  in Proc.
<1981 ACM Symp. on Symbolic and Algebraic Computation  (editor P. S.
<Wang), pp. 44-50.
<
<M. J. Prelle and M. F. Singer,  "Elementary first integrals of DEs,"
<Trans. AMS 279 (1983), 215-229. Kovacic's algorithm has been
<implemented in MACSYMA and MAPLE.
<
<See Saunders above for MACSYMA and the following for MAPLE. 
<
<B. W. Char, G. J. Fee, K. O. Geddes, G. H. Gonnet, and M. B. Monagan,
<"A tutorial introduction to MAPLE,"   J. Symbolic Computation 2 (1986),
<179-200.
<
<C. Smith,   "A discussion and implementation of Kovacic's algorithm for
<ODEs,"   University of Waterloo Computer Science Dept. Research Report
<CS-84-35 (1984).
-- 
Tim Chow     tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949