Gompertz constant

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

It can be defined by the continued fraction

${\displaystyle G={\frac {1}{2-{\frac {1}{4-{\frac {4}{6-{\frac {9}{8-{\frac {16}{10-{\frac {25}{12-{\frac {36}{14-{\frac {49}{16-\dots }}}}}}}}}}}}}}}},}$

or, alternatively, by

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

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

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

The numerical value of ${\displaystyle G}$ is about

${\displaystyle G=0.596347362323194074341078499369279376074\dots }$

During the studying divergent infinite series Euler met with ${\displaystyle G}$ via, for example, the above integral representations. Le Lionnais called ${\displaystyle G}$ as Gompertz constant by its role in survival analysis.[1]

Identities involving the Gompertz constant

The constant ${\displaystyle G}$ can be expressed by the exponential integral as

${\displaystyle G=-e{\textrm {Ei}}(-1).}$

Applying the Taylor expansion of ${\displaystyle {\textrm {Ei}}}$ we have that

${\displaystyle G=-e\left(\gamma +\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n\cdot n!}}\right).}$

Gompertz's constant is connected to the Gregory coefficients via the 2013 formula of I. Mező:[2]

${\displaystyle G=\sum _{n=0}^{\infty }{\frac {\ln(n+1)}{n!}}-\sum _{n=0}^{\infty }C_{n+1}\{e\cdot n!\}-{\frac {1}{2}}.}$