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

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

G={\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 $G$ is in the following integrals:

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

G=0.596347362323194074341078499369279376074\dots

When Euler studied divergent infinite series, he encountered $G$ via, for example, the above integral representations. Le Lionnais called $G$ the Gompertz constant because of its role in survival analysis.[1]

Identities involving the Gompertz constant

The constant $G$ can be expressed by the exponential integral as

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

Applying the Taylor expansion of ${\textrm {Ei}}$ we have the series representation

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]

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

Notes

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

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[Finch-1] Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. pp. 425–426.

[Mezo-2] Mező, István (2013). "Gompertz constant, Gregory coefficients and a series of the logarithm function". Journal of Analysis and Number Theory (7): 1–4.