Gompertz constant

In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by ${\displaystyle \delta }$, appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz.

It can be defined by the continued fraction

${\displaystyle \delta ={\frac {1}{2-{\frac {1}{4-{\frac {4}{6-{\frac {9}{{8-\qquad \qquad } \atop {\qquad }{}^{\ddots }{-{\frac {n^{2}}{2n+2-\dots }}}}}}}}}}},}$

or, alternatively, by

${\displaystyle \delta =1-{\frac {1}{3-{\frac {2}{5-{\frac {6}{7-{\frac {12}{{9-\qquad \qquad } \atop {\qquad }{}^{\ddots }{-{\frac {n(n+1)}{2n+3-\dots }}}}}}}}}}}}$

or

${\displaystyle \delta ={\frac {1}{1+{\frac {1}{1+{\frac {1}{1+{\frac {2}{1+{\frac {2}{1+{\frac {3}{1+{\frac {3}{1+{\frac {4}{1+\dots }}}}}}}}}}}}}}}}.}$

The most frequent appearance of ${\displaystyle \delta }$ is in the following integrals:

${\displaystyle \delta =\int _{0}^{\infty }\ln(1+x)e^{-x}dx=\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}dx=\int _{0}^{1}{\frac {1}{1-\ln(x)}}dx.}$

The first integral defines ${\displaystyle \delta }$, and the second and third follow from an integration of parts and a variable substitution respectively. The numerical value of ${\displaystyle \delta }$ is about

${\displaystyle \delta =0.596347362323194074341078499369279376074\dots }$

When Euler studied divergent infinite series, he encountered ${\displaystyle \delta }$ via, for example, the above integral representations. Le Lionnais called ${\displaystyle \delta }$ the Gompertz constant because of its role in survival analysis. The summation of negative integral values in gamma function with alternative negative signs upto infinity yields Euler Gompertz Constant. Γ(0) - Γ(-1) + Γ(-2) - Γ(-3) +...... = ${\displaystyle \delta =0.596347362323194074341078499369279376074\dots }$

In 2009 Alexander Aptekarev proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational. This result was improved in 2012 by Tanguy Rivoal where he proved that at least one of them is transcendental.