*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Let be Chebyshev Constants. Schönhage (1973) proved that

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

**References**

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/onenin/onenin.html

Carpenter, A. J.; Ruttan, A.; and Varga, R. S. ``Extended Numerical Computations on the `' Conjecture in Rational Approximation Theory.''
In *Rational approximation and interpolation (Tampa, Fla., 1983)* (Ed. P. R. Graves-Morris, E. B. Saff, and R. S. Varga).
New York: Springer-Verlag, pp. 383-411, 1984.

Cody, W. J.; Meinardus, G.; and Varga, R. S. ``Chebyshev Rational Approximations to in and Applications to
Heat-Conduction Problems.'' *J. Approx. Th.* **2**, 50-65, 1969.

Dunham, C. B. and Taylor, G. D. ``Continuity of Best Reciprocal Polynomial Approximation on .''
*J. Approx. Th.* **30**, 71-79, 1980.

Gonchar, A. A. ``Rational Approximations of Analytic Functions.'' *Amer. Math. Soc. Transl. Ser. 2* **147**, 25-34, 1990.

Gonchar, A. A. and Rakhmanov, E. A. ``Equilibrium Distributions and Degree of Rational Approximation of Analytic
Functions.'' *Math. USSR Sbornik* **62**, 305-348, 1980.

Magnus, A. P. ``On Freud's Equations for Exponential Weights, Papers Dedicated to the Memory of Géza Freud.''
*J. Approx. Th.* **46**, 65-99, 1986.

Rahman, Q. I. and Schmeisser, G. ``Rational Approximation to the Exponential Function.'' In *Padé and Rational Approximation,
(Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976)* (Ed. E. B. Saff and R. S. Varga).
New York: Academic Press, pp. 189-194, 1977.

Schönhage, A. ``Zur rationalen Approximierbarkeit von über .'' *J. Approx. Th.* **7**, 395-398, 1973.

Varga, R. S. *Scientific Computations on Mathematical Problems and Conjectures.* Philadelphia, PA: SIAM, 1990.

© 1996-9

1999-05-26