Gompertz constant

In mathematics, the Gompertz constant or Euler–Gompertz constant,^[1]^[2] denoted by $\delta$ , appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz.

It can be defined via the exponential integral as:^[3]

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

The numerical value of $\delta$ is about

δ

= 0.596347362323194074341078499369... (sequence A073003 in the OEIS).

History

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

In 1962, Shidlovski proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational.^[4] This result was improved in 2012 by Tanguy Rivoal, who proved that at least one of them is transcendental.^[2]^[4]^[5]^[6]

Identities involving the Gompertz constant

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

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

which follow from the definition of $δ$ by integration of parts and a variable substitution respectively.

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

\delta =-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 following formula:^[7]

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

The Gompertz constant also happens to be the regularized value of the summation of alternating factorials of all natural numbers (1 − 1 + 2 − 6 + 24 − 120 + ⋯), which is defined by Borel summation:^[2]

\delta =\sum _{k=0}^{\infty }(-1)^{k}k!

It is also related to several polynomial continued fractions:^[1]^[2]

{\frac {1}{\delta }}=2-{\cfrac {1^{2}}{4-{\cfrac {2^{2}}{6-{\cfrac {3^{2}}{8-{\cfrac {4^{2}}{\ddots {\cfrac {n^{2}}{2(n+1)-\dots }}}}}}}}}}

{\frac {1}{\delta }}=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {2}{1+{\cfrac {2}{1+{\cfrac {3}{1+{\cfrac {3}{1+{\cfrac {4}{\dots }}}}}}}}}}}}}}

{\frac {1}{1-\delta }}=3-{\cfrac {2}{5-{\cfrac {6}{7-{\cfrac {12}{9-{\cfrac {20}{\ddots {\cfrac {n(n+1)}{2n+3-\dots }}}}}}}}}}

Notes

1 2 3 Finch, Steven R. (2003). Mathematical Constants. Encyclopedia of Mathematics and Its Applications. Vol. 94. Cambridge University Press. pp. 425–426. ISBN 978-0-521-81805-6.
1 2 3 4 Lagarias, Jeffrey C. (2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv:1303.1856. doi:10.1090/S0273-0979-2013-01423-X.
↑ Weisstein, Eric W. "Gompertz Constant". MathWorld. Retrieved 1 August 2026.
1 2 Aptekarev, A. I. (2009). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].
↑ Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525.
↑ Waldschmidt, Michel (2023). "On Euler's Constant" (PDF). Sorbonne Université, Institut de Mathématiques de Jussieu, Paris.
↑ Mező, István (2014). "Gompertz constant, Gregory coefficients and a series of the logarithm function" (PDF). Journal of Analysis and Number Theory. 2 (2): 33–36.

External links

[Finch-1] 1 2 3 Finch, Steven R. (2003). Mathematical Constants. Encyclopedia of Mathematics and Its Applications. Vol. 94. Cambridge University Press. pp. 425–426. ISBN 978-0-521-81805-6.

[:0-2] 1 2 3 4 Lagarias, Jeffrey C. (2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv:1303.1856. doi:10.1090/S0273-0979-2013-01423-X.

[3] Weisstein, Eric W. "Gompertz Constant". MathWorld. Retrieved 1 August 2026.

[Aptekarev2009-4] 1 2 Aptekarev, A. I. (2009). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].

[5] Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525.

[6] Waldschmidt, Michel (2023). "On Euler's Constant" (PDF). Sorbonne Université, Institut de Mathématiques de Jussieu, Paris.

[Mezo-7] Mező, István (2014). "Gompertz constant, Gregory coefficients and a series of the logarithm function" (PDF). Journal of Analysis and Number Theory. 2 (2): 33–36.

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History

Identities involving the Gompertz constant

Notes

External links