Gompertz constant
In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by [math]\displaystyle{ \delta }[/math], appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz. It can be defined by the continued fraction
- [math]\displaystyle{ \delta = \frac{1}{2-\frac{1}{4-\frac{4}{6-\frac{9}{{8-\qquad\qquad}\atop{\qquad }{}^\ddots {-\frac{n^2}{2n+2-\dots}}}}}} , }[/math]
or, alternatively, by
- [math]\displaystyle{ \delta = 1-\frac{1}{3-\frac{2}{5-\frac{6}{7-\frac{12}{{9-\qquad\qquad}\atop{\qquad }{}^\ddots {-\frac{n(n+1)}{2n+3-\dots}}}}}} }[/math]
or
- [math]\displaystyle{ \delta = \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}}}}}}}}. }[/math]
The most frequent appearance of [math]\displaystyle{ \delta }[/math] is in the following integrals:
- [math]\displaystyle{ \delta = \int_0^\infty\ln(1+x)e^{-x}dx=\int_0^\infty\frac{e^{-x}}{1+x}dx=\int_0^1\frac{1}{1-\ln(x)}dx. }[/math]
The first integral defines [math]\displaystyle{ \delta }[/math], and the second and third follow from an integration of parts and a variable substitution respectively. The numerical value of [math]\displaystyle{ \delta }[/math] is about
- [math]\displaystyle{ \delta = 0.596347362323194074341078499369279376074\dots }[/math]
When Euler studied divergent infinite series, he encountered [math]\displaystyle{ \delta }[/math] via, for example, the above integral representations. Le Lionnais called [math]\displaystyle{ \delta }[/math] the Gompertz constant because of its role in survival analysis.[1] The summation of negative integral values in gamma function with alternative negative signs upto infinity yields Euler Gompertz Constant. Γ(0) - Γ(-1) + Γ(-2) - Γ(-3) +...... = [math]\displaystyle{ \delta = 0.596347362323194074341078499369279376074\dots }[/math]
In 2009 Alexander Aptekarev proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational. This result was improved in 2012 by Tanguy Rivoal where he proved that at least one of them is transcendental.[2][3][4]
Identities involving the Gompertz constant
The constant [math]\displaystyle{ \delta }[/math] can be expressed by the exponential integral as
- [math]\displaystyle{ \delta = -e\operatorname{Ei}(-1). }[/math]
Applying the Taylor expansion of [math]\displaystyle{ \operatorname{Ei} }[/math] we have the series representation
- [math]\displaystyle{ \delta = -e\left(\gamma+\sum_{n=1}^\infty\frac{(-1)^n}{n\cdot n!}\right). }[/math]
Gompertz's constant is connected to the Gregory coefficients via the 2013 formula of I. Mező:[5]
- [math]\displaystyle{ \delta = \sum_{n=0}^\infty\frac{\ln(n+1)}{n!}-\sum_{n=0}^\infty C_{n+1}\{e\cdot n!\}-\frac{1}{2}. }[/math]
Sum of alternating factorials
The Gompertz constant also happens to be the regularized value[dubious ] of the following divergent series:
- [math]\displaystyle{ \sum_{k=0}^{\infty} (-1)^k k! = 1 - 1 + 2 - 6 + 24 - 120 + \ldots }[/math]
Notes
- ↑ Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. pp. 425–426.
- ↑ Aptekarev, A. I. (2009-02-28). "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" (in EN). Michigan Mathematical Journal 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285. https://projecteuclid.org/euclid.mmj/1339011525.
- ↑ Lagarias, Jeffrey C. (2013-07-19). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society 50 (4): 527–628. doi:10.1090/S0273-0979-2013-01423-X. ISSN 0273-0979.
- ↑ Mező, István (2013). "Gompertz constant, Gregory coefficients and a series of the logarithm function". Journal of Analysis and Number Theory (7): 1–4. https://www.naturalspublishing.com/files/published/j18jp677r69ri8.pdf.
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