Special constant related to the exponential integral
In mathematics , the Gompertz constant or Euler–Gompertz constant ,[ 1] [ 2]
δ
{\displaystyle \delta }
special functions . It is named after Benjamin Gompertz .
It can be defined via the exponential integral as:[ 3]
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
{\displaystyle \delta }
δ = 347 362 323 194 074 341 078 499 369 A073003 OEIS ).When Euler studied divergent infinite series, he encountered
δ
{\displaystyle \delta }
Le Lionnais called
δ
{\displaystyle \delta }
survival analysis .[ 1]
In 1962, A. B. Shidlovski proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational .[ 4] Tanguy Rivoal where he proved that at least one of them is transcendental .[ 2] [ 5] [ 6] [ 7]
Identities involving the Gompertz constant
The most frequent appearance of
δ
{\displaystyle \delta }
δ
=
∫
0
∞
ln
(
1
+
x
)
e
−
x
d
x
{\displaystyle \delta =\int _{0}^{\infty }\ln(1+x)e^{-x}dx}
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
=
−
e
(
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:[ 8]
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 summation of alternating factorials of all natural numbers (1 − 1 + 2 − 6 + 24 − 120 + ⋯ ), which is defined by Borel summation :[ 2]
δ
=
∑
k
=
0
∞
(
−
1
)
k
k
!
{\displaystyle \delta =\sum _{k=0}^{\infty }(-1)^{k}k!}
It is also related to several polynomial continued fractions :[ 1] [ 2]
1
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
=
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
=
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. (28 February 2009). "On linear forms containing the Euler constant". arXiv :0902.1768 math.NT ]. ^ 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
