A transformation of rational functions
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.
scientific article; zbMATH DE number 2102322