The "One-Ninth" Constant

We are concerned here with rational approximation of exp(-x) on the half-line [0,). Let denote the error of best uniform approximation:

where

There are two cases of special interest, when m=0 and m=n, since clearly

Many researchers [1,2,10] have studied these constants , referred to as Chebyshev constants in [2]. We mention the work of only a few. Schönhage [11] proved that

which led several people to conjecture that

Numerical evidence uncovered by Schönhage [12] and Trefethen & Gutknecht [13] suggested that the conjecture is false. Carpenter, Ruttan and Varga [14] calculated the Chebyshev constants to an accuracy of 200 digits up to n = 30 and carefully obtained

although a proof that the limit even existed was still to be found.

Building on Opitz & Scherer [15] and Magnus [16], Gonchar & Rakhmanov [2,3] proved that the limit exists and that it equals

where K is the complete elliptic integral of the first kind

and the constant c is defined as follows. Let E be the complete elliptic integral of the second kind

(One's first encounter with K and E is often with regard to computing the period of a physical pendulum and finding the arclength of an ellipse.) The constant c is defined simply as the unique solution, 0 < c < 1, of the equation

The appearance of elliptic integrals here reminds one of work done over one hundred years before. In 1877, Zolotarev determined a number of exact solutions to approximation problems using elliptic functions [1] in research which was far ahead of its time. See Bernstein's constant for more details.

Gonchar and Rakhmanov's exact disproof of the "one-ninth" conjecture utilized ideas from complex potential theory, which seems far removed from the rational approximation of exp(-x)! They also obtained a number theoretic characterization of the "one-ninth" constant . If

where

then f is complex-analytic in the open unit disc. The unique positive root of the equation

is the constant . Another way of writing a_j is as follows [5]. If

is the prime factorization of the integer j, where each p_i is an odd prime, m ≥ 0 and m_i ≥ 1 , then

Carpenter [2] computed to 101 digits using this equation.

Here is another expression due to Magnus [16]. The one-ninth constant is the unique solution, with 0 < x < 1, of the equation

which turns out to have been studied one hundred years earlier by Halphen [4]. Halphen was interested in theta functions and computed to six digits, clearly unaware that this constant would become prominent a century later! Varga [2] has suggested that be renamed the Halphen constant. So many researchers have contributed to the solution of this approximation problem, however, that retaining the amusingly inaccurate "one-ninth" designation might be simplest.

Relevant Mathcad files will be included as time permits. Plouffe gave accurate approximations of and 1/ in the Inverse Symbolic Calculator web pages.

Postscript

The constant c=0.9089085575... defining

arises in a completely unrelated field: the study of Euler elasticae [6-8].

A quotient of elliptic functions, similar to that above, occurs in our essay on Grötzsch ring constants.

References

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A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree of rational approximation of analytic functions, Math. USSR Sbornik 62 (1989) 305-348; MR 89h:30054.
G.-H. Halphen, Traité des fonctions elliptiques et de leurs applications, v. 1, Gauthier-Villars, 1886, p. 287 (Cornell Digital Library).
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