The Fransén–Robinson constant has numerical value F = 2.8077702420285... (sequence A058655 in OEIS), with the continued fraction representation [2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, ...] (sequence A046943 in OEIS). Its proximity to Euler's number e = 2.71828... follows from the fact that the integral can be approximated by
the standard series for e. The difference is given by
and also by
The Fransén–Robinson constant can also be expressed using the Mittag-Leffler function as the limit
It is however unknown whether F can be expressed in closed form in terms of other known constants.
A fair amount of effort has been made to calculate the numerical value of the Fransén–Robinson constant with high accuracy. The value was computed to 36 decimal places by Herman P. Robinson using 11-point Newton–Cotes quadrature, with 65 digits by A. Fransén using Euler–Maclaurin summation, and with 80 digits by Fransén and S. Wrigge using Taylor series and other methods. William A. Johnson computed 300 digits, and Pascal Sebah was able to compute 600 digits using Clenshaw–Curtis integration.
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- Fransen, Arne; Wrigge, Staffan (1980). "High-Precision values of the Gamma function and of some related coefficients". Mathematics of Computation 34 (150): 553–566. doi:10.2307/2006104. MR 0559204.
- Fransen, Arne (1981). "Addendum and Corrigendum to "High-Precision values of the Gamma function and of some related coefficients"". Mathematics of Computation 37 (155): 233–235. doi:10.2307/2007517. MR 0616377.
- Finch, Steve. "Fransén–Robinson Constant".[dead link]
- Weisstein, Eric W., "Fransén–Robinson Constant", MathWorld.
- Borwein, Jonathan; David Bailey, Roland Girgensohn (2003). Experimentation in Mathematics – Computational Paths to Discovery. A K Peters. p. 288. ISBN 1-56881-136-5.
- Fransen, Arne. "The value of the Fransén–Robinson constant to 300 decimal places".