
The "OneNinth" Constant
We are concerned here with rational approximation of exp(x) on the
halfline [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 "oneninth" 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 "oneninth" constant
. If
where
then f is complexanalytic 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 oneninth 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 "oneninth" 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 [68].
A quotient of elliptic functions, similar to that above, occurs in our essay on
Grötzsch ring constants.
References
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Functions, Cambridge Univ. Press, 1987;
MR 89i:41022.
 R. S. Varga, Scientific Computation on Mathematical Problems and
Conjectures, SIAM, 1990;
MR 92b:65012.
 A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree
of rational approximation of analytic functions, Math. USSR Sbornik
62 (1989) 305348;
MR 89h:30054.
 G.H. Halphen, Traité
des fonctions elliptiques et de leurs applications,
v. 1, GauthierVillars, 1886, p. 287 (Cornell Digital Library).
 A. A. Gonchar, Rational approximations of analytic functions, Amer.
Math. Soc. Transl. Ser. 2, 147 (1990) 2534;
MR 89e:30066.
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MR 85f:01004.
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MR 39 #6536.
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SIAM J. Numer. Anal. 19 (1982) 10671080;
MR 83k:41016.
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MR 85g:41024.
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MR 88f:41027.
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constants, Nonlinear Numerical Methods and Rational Approximation, Reidel, 1988, pp. 105132,
MR 90j:65035.
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exponential function (the "1/9" problem) by the CarathéodoryFejér method, Nonlinear Numerical
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preprint;
MR 96b:41023.
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Numer. Algor. 24 (2000) 117139 ; preprint.
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