Gompertz constant

{{Short description|Special constant related to the exponential integral}}

In mathematics, the Gompertz constant or Euler–Gompertz constant,{{Cite journal |last=Lagarias |first=Jeffrey C. |date=2013-07-19 |title=Euler's constant: Euler's work and modern developments |journal=Bulletin of the American Mathematical Society |volume=50 |issue=4 |pages=527–628 |arxiv=1303.1856 |doi=10.1090/S0273-0979-2013-01423-X |issn=0273-0979 |s2cid=119612431}} 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:{{Cite web |last=Weisstein |first=Eric W. |title=Gompertz Constant |url=https://mathworld.wolfram.com/GompertzConstant.html |access-date=2024-10-20 |website=mathworld.wolfram.com |language=en}}

: $\delta = -e\operatorname{Ei}(-1)=\int_0^\infty\frac{e^{-x}}{1+x}dx.$

The numerical value of $\delta$ is about

:{{mvar|δ}} = {{val|0.596347362323194074341078499369}}... {{OEIS|id=A073003}}.

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.

In 1962, A. B. Shidlovski proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational.{{cite arXiv |last=Aptekarev |first=A. I. |date=28 February 2009 |title=On linear forms containing the Euler constant |class=math.NT |eprint=0902.1768}} This result was improved in 2012 by Tanguy Rivoal where he proved that at least one of them is transcendental.{{cite arXiv |eprint=0902.1768 |class=math.NT |first=A. I. |last=Aptekarev |title=On linear forms containing the Euler constant |date=2009-02-28}}{{Cite journal |last=Rivoal |first=Tanguy |date=2012 |title=On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant |url=https://projecteuclid.org/euclid.mmj/1339011525 |journal=Michigan Mathematical Journal |language=EN |volume=61 |issue=2 |pages=239–254 |doi=10.1307/mmj/1339011525 |issn=0026-2285 |doi-access=free}}{{Cite web |last=Waldschmidt |first=Michel |date=2023 |title=On Euler's Constant |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/EulerConstant.pdf |place=Sorbonne Université, Institut de Mathématiques de Jussieu, Paris}}

Identities involving the Gompertz constant

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

: $\delta = \int_0^\infty\ln(1+x)e^{-x}dx$

: $\delta = \int_0^1\frac{1}{1-\ln(x)}dx$

which follow from the definition of {{mvar|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 2013 formula of I. Mező:{{cite journal | last = Mező | first = István| title = Gompertz constant, Gregory coefficients and a series of the logarithm function | journal = Journal of Analysis and Number Theory | year = 2013 | issue = 7 | pages = 1–4 | url=https://www.naturalspublishing.com/files/published/j18jp677r69ri8.pdf}}

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

The Gompertz constant also happens to be the regularized value of the summation of alternating factorials of all positive integers and summing over all factorial values of every integer leads to zero {divergent series:{{dubious|date=November 2023}}}

: $\sum_{k=-\infty}^{\infty} (-1)^k k! = 0$

It is also related to several polynomial continued fractions:

: $\frac1\delta = 2-\cfrac{1^2}{4-\cfrac{2^2}{6-\cfrac{3^2}{8-\cfrac{4^2}{\ddots \cfrac{n^2}{2(n+1)-\dots}}}}}$

: $\frac1\delta = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{3}{1+\cfrac{3}{1+\cfrac{4}{\dots}}}}}}}$

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

Notes

{{reflist|refs=

{{Cite book

| last = Finch | first = Steven R.

| title = Mathematical Constants

| year = 2003

| publisher = Cambridge University Press

| pages = 425–426

}}

External links

[http://mathworld.wolfram.com/GompertzConstant.html Wolfram MathWorld]
[https://oeis.org/A073003 OEIS entry]

Category:Analysis

Category:Mathematical constants