View source for Fransén–Robinson constant

The '''Fransén–Robinson constant''', sometimes denoted ''F'', is the [[mathematical constant]] that represents the area between the graph of the [[reciprocal Gamma function]], {{math|1/&Gamma;(''x'')}}, and the positive ''x'' axis. That is,

:<math>F = \int_0^\infty \frac{1}{\Gamma(x)}\,dx = 2.8077702420285...</math>

==Other expressions==
The Fransén–Robinson constant has numerical value {{nowrap|''F'' {{=}} 2.8077702420285...}} {{OEIS|id=A058655}}, and [[continued fraction]] representation [2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, ...] {{OEIS|id=A046943}}. The constant is somewhat close to [[e (mathematical constant)|Euler's number]] {{nowrap|''e'' {{=}} 2.71828... .}}  This fact can be explained by approximating the integral by a sum:

:<math>F = \int_0^\infty \frac{1}{\Gamma(x)}\,dx \approx \sum_{n=1}^\infty \frac{1}{\Gamma(n)} = \sum_{n=0}^\infty \frac{1}{n!},</math>

and this sum is the standard series for ''e''. The difference is

:<math>F - e = \int_0^\infty \frac{e^{-x}}{\pi^2 + (\ln x)^2}\,dx</math>

or equivalently

:<math>F = e + \frac{1}{\pi} \int_{-\pi/2}^{\pi/2} e^{\pi \tan \theta} e^{-e^{\pi \tan \theta}}\,d\theta.</math>

The Fransén–Robinson constant can also be expressed using the [[Mittag-Leffler function]] as the limit

:<math>F = \lim_{\alpha \to 0} \alpha E_{\alpha, 0}(1).</math>

It is however unknown whether ''F'' can be expressed in [[closed-form expression|closed form]] in terms of other known constants.

==Calculation history==
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&nbsp;decimal places by Herman P. Robinson using 11&nbsp;point [[Newton–Cotes quadrature]], to 65&nbsp;digits by A. Fransén using [[Euler–Maclaurin summation]], and to 80&nbsp;digits by Fransén and S. Wrigge using [[Taylor series]] and other methods. William A. Johnson computed 300&nbsp;digits, and Pascal Sebah was able to compute 600&nbsp;digits using [[Clenshaw–Curtis quadrature|Clenshaw–Curtis integration]].{{citation needed|date=June 2019}}

==References==
* {{cite journal |first1=Arne |last1=Fransen |title=Accurate determination of the inverse Gamma integral |journal=BIT |volume=19 |issue=1 |pages=137–138 |mr=0530126 |doi=10.1007/BF01931232 |year=1979}}

* {{cite journal |first1=Arne |last1=Fransen |first2=Staffan |last2=Wrigge |title=High-Precision values of the Gamma function and of some related coefficients |journal=Mathematics of Computation |volume=34 |issue=150 |pages=553–566 |year=1980 |mr=0559204 |doi=10.2307/2006104}}

* {{cite journal |first1=Arne |last1=Fransen |title=Addendum and corrigendum to "High-Precision values of the Gamma function and of some related coefficients" |journal=[[Mathematics of Computation]] |volume=37 |issue=155 |pages=233–235 |year=1981 |mr=0616377 |doi=10.2307/2007517}}

* {{cite web |first1=Steve |last1=Finch |url=http://www.mathsoft.com/mathsoft_resourcesmathsoft_constants/Number_Theory_Constants/2030.asp |title=Fransén–Robinson Constant |url-status=dead}} {{dead link|date=May 2010}}

* {{MathWorld|urlname=Fransen-RobinsonConstant|title=Fransén–Robinson Constant}}

* {{cite book |last1=Borwein |first1=Jonathan |first2=David |last2=Bailey |first3=Roland |last3=Girgensohn |title=Experimentation in Mathematics – Computational Paths to Discovery |publisher=A. K. Peters |year=2003 |isbn=1-56881-136-5 |page=288}}

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[[Category:Mathematical constants]]
[[Category:Gamma and related functions]]


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