Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Rational Landen transformations on $ \mathbb{R}$

Author(s): Dante Manna; Victor H. Moll.
Journal: Math. Comp. 76 (2007), 2023-2043.
MSC (2000): Primary 33F05; Secondary 26C15
Posted: May 3, 2007
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: The Landen transformation $ (a,b) \mapsto ( (a+b)/2,\sqrt{ab} )$ preserves the value of an elliptic integral, and its iteration produces the classical arithmetic-geometric mean AGM$ (a,b)$. We present analogous transformations for rational functions integrated over the whole real line.

References:

1.
G. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, New York, 1999. MR 1688958 (2000g:33001)

2.
M. Artin, Algebra, Prentice Hall, Englewood Cliffs, New Jersey, 1991. MR 1129886 (92g:00001)

3.
G. Boros, O. Espinosa, and V. Moll, On some families of integrals solvable in terms of polygamma and negapolygamma functions, Integrals Transforms and Special Functions 14 (2003), 187-203. MR 1982825 (2004f:33004)

4.
G. Boros and V. Moll, A rational Landen transformation. The case of degree $ 6$, Contemporay Mathematics. Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis (Knopp G., Mendoza E.T., Quinto E. L., Grinberg S., Berhanu M., eds.), vol. 251, American Mathematical Society, 2000, pp. 83-89. MR 1771260 (2001h:33029)

5.
-, Landen transformation and the integration of rational functions, Math. Comp. 71 (2001), 649-668. MR 1885619 (2004c:33044)

6.
J. M. Borwein and P. B. Borwein, Pi and the AGM. A study in analytic number theory and computational complexity, 1st ed., Wiley, New York, 1987. MR 877728 (89a:11134)

7.
M. Bronstein, Symbolic integration. I. Transcendental functions, Algorithms and Computation in Mathematics, vol. 1, Springer-Verlag, 1997. MR 1430096 (98a:68104)

8.
M. Chamberland and V. Moll, Dynamics of the degree six Landen transformation, Discrete and Dynamical Systems 15 (2006), no. 3, 905-919. MR 2220755 (2006m:39012)

9.
K. F. Gauss, Arithmetisch Geometrisches Mittel, Werke 3 (1799), 361-432.

10.
-, Disquisitiones generales circa seriem infinitam, Werke 3 (1812), 123-162.

11.
K. Geddes, S. R. Czapor, and G. Labahn, Algorithms for computer algebra, Kluwer, Dordrecht, The Netherlands, 1992. MR 1256483 (96a:68049)

12.
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Edited by A. Jeffrey and D. Zwillinger. Academic Press, New York, 2000. MR 1773820 (2001c:00002)

13.
C. Hermite, Sur l'integration des fractions rationelles, Nouvelles Annales de Mathematiques ($ 2^{eme}$ serie) 11 (1872), 145-148.

14.
E. Horowitz, Algorithms for partial fraction decomposition and rational function integration, Proc. of SYMSAM'71, ACM Press (1971), 441-457.

15.
J. Hubbard and V. Moll, A geometric view of rational Landen transformation, Bull. London Math. Soc. 35 (2003), 293-301. MR 1960939 (2003k:33028)

16.
D. Lazard and R. Rioboo, Integration of rational functions: rational computation of the logarithmic part, Journal of Symbolic Computation 9 (1990), 113-116. MR 1056840 (91h:68091)

17.
D. Manna and V. Moll, A simple example of a new class of Landen transformations, American Mathematical Monthly 114 (2007), 232-241. MR 2290287

18.
H. McKean and V. Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, New York, 1997. MR 1471703 (98g:14032)

19.
M. W. Ostrogradsky, De l'integration des fractions rationelles, Bulletin de la Classe Physico-Mathematiques de l'Academie Imperiale des Sciences de St. Petersbourgh IV (1845), 145-167, 286-300.

20.
J. F. Ritt, Permutable rational functions, Trans. Amer. Math. Soc. 25 (1923), 399-448. MR 1501252

21.
M. Rothstein, A new algorithm for the integration of exponential and logarithmic functions, Proc. of the $ 1977$ MACSYMA Users Conference, NASA Pub., CP-2012 (1977), 263-274.

22.
B. M. Trager, Algebraic factoring and rational function integration, Proc. SYMSAC (1976), 219-226.

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 33F05, 26C15

Retrieve articles in all Journals with MSC (2000): 33F05, 26C15

Additional Information:

Dante Manna
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisianna 70118
Address at time of publication: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Email: dmanna@mathstat.dal.ca

Victor H. Moll
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisianna 70118
Email: vhm@math.tulane.edu

DOI: 10.1090/S0025-5718-07-01954-0
PII: S 0025-5718(07)01954-0
Keywords: Integrals, transformations
Received by editor(s): November 2, 2005
Posted: May 3, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google