Difference between revisions of "Fransén–Robinson constant"

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The Fransén–Robinson constant is defined to be the number $F$ given by the formula
 
The Fransén–Robinson constant is defined to be the number $F$ given by the formula
$$F = \displaystyle\int_0^{\infty} \dfrac{1}{\Gamma(x)} dx.$$
+
$$F = \displaystyle\int_0^{\infty} \dfrac{1}{\Gamma(x)} dx,$$
 +
where $\dfrac{1}{\Gamma}$ denotes the [[reciprocal gamma function]].
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Proposition (Relation to [[e|$e$]] and [[pi|$\pi$]]):</strong> $F=e+\displaystyle\int_0^{\infty} \dfrac{e^{-x}}{\pi^2+\log(x)^2}.$
 
<strong>Proposition (Relation to [[e|$e$]] and [[pi|$\pi$]]):</strong> $F=e+\displaystyle\int_0^{\infty} \dfrac{e^{-x}}{\pi^2+\log(x)^2}.$

Revision as of 18:33, 7 February 2016

The Fransén–Robinson constant is defined to be the number $F$ given by the formula $$F = \displaystyle\int_0^{\infty} \dfrac{1}{\Gamma(x)} dx,$$ where $\dfrac{1}{\Gamma}$ denotes the reciprocal gamma function.

Proposition (Relation to $e$ and $\pi$): $F=e+\displaystyle\int_0^{\infty} \dfrac{e^{-x}}{\pi^2+\log(x)^2}.$

Proof: proof goes here █