Special constant related to the exponential integral
In mathematics , the Gompertz constant or Euler–Gompertz constant ,[ 1] [ 2]
δ δ -->
{\displaystyle \delta }
special functions . It is named after Benjamin Gompertz .
It can be defined via the exponential integral as:[ 3]
δ δ -->
=
− − -->
e
Ei
-->
(
− − -->
1
)
=
∫ ∫ -->
0
∞ ∞ -->
e
− − -->
x
1
+
x
d
x
.
{\displaystyle \delta =-e\operatorname {Ei} (-1)=\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}dx.}
The numerical value of
δ δ -->
{\displaystyle \delta }
δ = 347 362 323 194 074 341 078 499 369 A073003 OEIS ).When Euler studied divergent infinite series, he encountered
δ δ -->
{\displaystyle \delta }
Le Lionnais called
δ δ -->
{\displaystyle \delta }
survival analysis .[ 1]
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] [ 4] [ 5] [ 6]
Identities involving the Gompertz constant
The most frequent appearance of
δ δ -->
{\displaystyle \delta }
δ δ -->
=
∫ ∫ -->
0
∞ ∞ -->
ln
-->
(
1
+
x
)
e
− − -->
x
d
x
{\displaystyle \delta =\int _{0}^{\infty }\ln(1+x)e^{-x}dx}
δ δ -->
=
∫ ∫ -->
0
1
1
1
− − -->
ln
-->
(
x
)
d
x
{\displaystyle \delta =\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
Ei
{\displaystyle \operatorname {Ei} }
δ δ -->
=
− − -->
e
(
γ γ -->
+
∑ ∑ -->
n
=
1
∞ ∞ -->
(
− − -->
1
)
n
n
⋅ ⋅ -->
n
!
)
.
{\displaystyle \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:[ 7]
δ δ -->
=
∑ ∑ -->
n
=
0
∞ ∞ -->
ln
-->
(
n
+
1
)
n
!
− − -->
∑ ∑ -->
n
=
0
∞ ∞ -->
C
n
+
1
{
e
⋅ ⋅ -->
n
!
}
− − -->
1
2
.
{\displaystyle \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 following divergent series :[ 2] [dubious discuss
∑ ∑ -->
k
=
0
∞ ∞ -->
(
− − -->
1
)
k
k
!
=
1
− − -->
1
+
2
− − -->
6
+
24
− − -->
120
+
… … -->
{\displaystyle \sum _{k=0}^{\infty }(-1)^{k}k!=1-1+2-6+24-120+\ldots }
It is also related to several polynomial continued fractions :[ 1] [ 2]
1
δ δ -->
=
2
− − -->
1
2
4
− − -->
2
2
6
− − -->
3
2
8
− − -->
4
2
⋱ ⋱ -->
n
2
2
(
n
+
1
)
− − -->
… … -->
{\displaystyle {\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 }}}}}}}}}}}
1
δ δ -->
=
1
+
1
1
+
1
1
+
2
1
+
2
1
+
3
1
+
3
1
+
4
… … -->
{\displaystyle {\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 }}}}}}}}}}}}}}}
1
1
− − -->
δ δ -->
=
3
− − -->
2
5
− − -->
6
7
− − -->
12
9
− − -->
20
⋱ ⋱ -->
n
(
n
+
1
)
2
n
+
3
− − -->
… … -->
{\displaystyle {\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
^ a b c
Finch, Steven R. (2003). Mathematical Constants . Cambridge University Press. pp. 425– 426.
^ a b c d 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. arXiv :1303.1856 doi :10.1090/S0273-0979-2013-01423-X . ISSN 0273-0979 . S2CID 119612431 . ^ Weisstein, Eric W. "Gompertz Constant" . mathworld.wolfram.com . Retrieved 2024-10-20 . ^ 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" . Michigan Mathematical Journal . 61 (2): 239– 254. doi :10.1307/mmj/1339011525 ISSN 0026-2285 . ^ Waldschmidt, Michel (2023). "On Euler's Constant" (PDF) . Sorbonne Université, Institut de Mathématiques de Jussieu, Paris. ^ Mező, István (2013). "Gompertz constant, Gregory coefficients and a series of the logarithm function" (PDF) . Journal of Analysis and Number Theory (7): 1– 4.
External links
