A transformation of rational functions
type: Item
instance of: scholarly article
title: A transformation of rational functions
zbMATH Open document ID: 1061.33018
author: George Boros
author: Michael Joyce
author: Victor H. Moll
published in: Elemente der Mathematik
publication date: 2004-09-22T00:00:00Z
review text: Let \(r(\theta)= \sqrt{a^2 \cos^2{\theta} + b^2 \sin^2{\theta}}\) for \(a,b >0, \theta \in (0, \pi/2),\) and let \[ I(a,b)= \int_0^{\pi/2} d{\theta}/ r(\theta). \] In the special case \(a=b>0\) we have \(I(a,a)= \pi/(2 a) \, ,\) and it is a remarkable fact that in the general case one may use this special case and the notion of the arithmetic geometric mean to obtain an elegant algorithm for \(I(a,b) \,.\) If \(a_0=a,\,b_0=b, a_{n+1}=(a_n+b_n)/2, b_{n+1}= \sqrt{a_n b_n} , n= 0,1,2,3, \dots ,\) then both sequences converge quadratically to the common limit, the arithmetic geometric mean \(AGM(a,b)\) of \(a\) and \(b.\) In the 1770's J. Landen proved that \(I(a,b)=I(a_1,b_1)=I(a_n,b_n)\) for all \(n=2,3,\dots .\) Then clearly \(I(a,b)=\pi/(2 AGM(a,b))\) (cf. the aforementioned special case \(a=b>0\,.\)) Thus the computation of the integral \(I(a,b)\) is reduced to the simple task of the computation of the \(AGM(a,b).\) For a survey of the AGM and its applications, see \textit{J. Arazy, T. Claesson, S. Janson} and \textit{J. Peetre}, [Proc. 19th Nordic Congr. Math., Reykjavik/Iceland 1984, 191-212 (1985; Zbl 0606.26007)] and \textit{J. Wimp} [Computation with recurrence relations. (Applicable Mathematics Series. Boston-London-Melbourne: Pitman Advanced Publishing Program) (1984; Zbl 0543.65084)]. Motivated by the example of the AGM, the authors consider integrals of the form \(\int_0^{\infty}(p(x)/q(x)) dx \) where \(p\) and \(q\) are polynomials and introduce in this context a transformation which they use for the study of the integral, thus extending some of their earlier work.
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 33C75
zbMATH DE Number: 2102322
zbMATH Keywords: Landen transformation
zbMATH Keywords: arithmetic-geometric mean
reviewed by: M. Vuorinen
MaRDI profile type: Publication
full work available at URL: s000170300010
OpenAlex ID: W2100697027
cites work: The double square root, Jacobi polynomials and Ramanujan's master theorem
cites work: A map on the space of rational functions
cites work: Q4826644
cites work: Artin's conjecture for primitive roots
cites work: The dynamical system generated by the \(3n+1\) function
DOI: 10.1007/S000170300010
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description: scientific article; zbMATH DE number 2102322