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
There are two cases of special interest, when m=0 and m=n,
Many researchers [1,2,10] have studied these constants
, referred to as Chebyshev constants
in . We mention the work of only a few. Schönhage  proved
which led several people to conjecture that
Numerical evidence uncovered by Schönhage  and Trefethen &
Gutknecht  suggested that the conjecture is false. Carpenter,
Ruttan and Varga  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
Building on Opitz & Scherer  and Magnus , 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 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  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
then f is complex-analytic in the open unit disc. The unique positive
root of the equation
is the constant . Another way
is as follows . If
is the prime factorization of the integer j, where each
is an odd prime,
mi ≥ 1
Carpenter  computed to
101 digits using this equation.
Here is another expression due to Magnus . 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 . Halphen was interested in theta functions and computed
to six digits, clearly unaware
that this constant would become prominent a century later! Varga  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
Symbolic Calculator web pages.
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.
- P. P. Petrushev and V. A. Popov, Rational Approximation of Real
Functions, Cambridge Univ. Press, 1987;
- R. S. Varga, Scientific Computation on Mathematical Problems and
Conjectures, SIAM, 1990;
- A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree
of rational approximation of analytic functions, Math. USSR Sbornik
62 (1989) 305-348;
- G.-H. Halphen, Traité
des fonctions elliptiques et de leurs applications,
v. 1, Gauthier-Villars, 1886, p. 287 (Cornell Digital Library).
- A. A. Gonchar, Rational approximations of analytic functions, Amer.
Math. Soc. Transl. Ser. 2, 147 (1990) 25-34;
- D. A. Singer, Curves whose curvature depends on distance from the origin,
Amer. Math. Monthly 106 (1999) 835-841;
- T. A. Ivey and D. A. Singer, Knot types, homotopies and stability of closed
elastic rods, Proc. London Math. Soc. 79 (1999) 429-450;
- C. Truesdell, The influence of elasticity on analysis: The classic heritage,
Bull. Amer. Math. Soc. 9 (1983) 293-310;
- R. S. Varga, Topics in Polynomial and Rational Interpolation and Approximation,
Les Presses de l'Université de Montréal, 1982;
- W. J. Cody, G. Meinardus and R. S. Varga, Chebyshev rational approximations to e-x in
[0,+) and applications to heat-conduction problems,
J. Approx. Theory 2 (1969) 50-65;
MR 39 #6536.
- A. Schönhage, Zur rationalen Approximierbarkeit von e-x über
[0,), J. Approx. Theory 7 (1973) 395-398;
MR 49 #3391.
- A. Schönhage, Rational approximation to e-x and related L2-problems,
SIAM J. Numer. Anal. 19 (1982) 1067-1080;
- L. N. Trefethen and M. H. Gutknecht, The Carathéodory-FejÚr method for real rational approximation,
SIAM J. Numer. Anal. 20 (1983) 420-436;
- A. J. Carpenter, A. Ruttan, and R. S. Varga, Extended numerical computations on the "1/9" conjecture in rational
approximation theory, Rational Approximation and Interpolation, Lecture Notes in Math. 1105,
Springer-Verlag, 1984, pp. 383-411.
- H.-U. Opitz and K. Scherer, On the rational approximation of e-x on [0,),
Constr. Approx. 1 (1985) 195-216;
- A. P. Magnus, On the use of the Carathéodory-Fejér method for investigating "1/9" and similar
constants, Nonlinear Numerical Methods and Rational Approximation, Reidel, 1988, pp. 105-132,
- A. P. Magnus, Asymptotics and super asymptotics for best rational approximation error norms to the
exponential function (the "1/9" problem) by the Carathéodory-Fejér method, Nonlinear Numerical
Methods and Rational Approximation II, Kluwer, 1994, pp. 173-185;
- A. P. Magnus and J. Meinguet, The elliptic functions and integrals of the "1/9" problem,
Numer. Algor. 24 (2000) 117-139 ; preprint.
Return to the Favorite Mathematical Constants main essay.
Copyright © 1995-2001 by Steven Finch.
All rights reserved.