Gompertz constant

In mathematics, the Gompertz constant or Euler-Gompertz constant, denoted by $G$, 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 $G$ 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 $G$ is about

${\displaystyle G=0.596347362323194074341078499369279376074\dots }$

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

Identities involving the Gompertz constant

The constant $G$ 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]

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

External links

• Wolfram MathWorld

Notes

1. Steven R., Finch (2003). Mathematical Constants. Cambridge University Press. pp. 425–426.
2. Mező, István (2013). "Gompertz constant, Gregory coefficients and a series of the logarithm function". Journal of Analysis and Number Theory (7): 1–4.