“Fransén–Robinson constant”的意思、由来-开放百科全书

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Fransén–Robinson constant

释义

References

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

The Fransén–Robinson constant has numerical value {{nowrap|F {{=}} 2.8077702420285...}} {{OEIS|id=A058655}} and the 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 Euler's number {{nowrap|e {{=}} 2.71828...}} This fact can be explained by approximating the integral by a sum:

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

or equivalently

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, to 65 digits by A. Fransén using Euler–Maclaurin summation, and to 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.

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_resources/mathsoft_constants/Number_Theory_Constants/2030.asp
|title=Fransén–Robinson Constant

}} {{dead link|date=May 2010}}

{{MathWorld|urlname=Fransen-RobinsonConstant|title=Fransén–Robinson Constant}}
{{cite book |last= Borwein |first= Jonathan |author2=David Bailey |author3=Roland Girgensohn |title= Experimentation in Mathematics – Computational Paths to Discovery |publisher= A K Peters |year= 2003 |isbn= 1-56881-136-5 |page= 288}}

2 : Mathematical constants|Gamma and related functions

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