In mathematics, the Gompertz constant or Euler–Gompertz constant,[1] denoted by
\delta
It can be defined via the exponential integral as:[2]
\delta=
| ||||
| -e\operatorname{Ei}(-1)=\int | ||||
| 0 |
dx.
The numerical value of
\delta
= ... .
When Euler studied divergent infinite series, he encountered
\delta
\delta
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.[3] [4] [5]
The most frequent appearance of
\delta
\delta=
| inftyln(1+x)e | |
| \int | |
| 0 |
-xdx
\delta=
| ||||
| \int | ||||
| 0 |
dx
which follow from the definition of by integration of parts and a variable substitution respectively.
Applying the Taylor expansion of
\operatorname{Ei}
\delta=
| ||||
| -e\left(\gamma+\sum | ||||
| n=1 |
\right).
Gompertz's constant is connected to the Gregory coefficients via the 2013 formula of I. Mező:[6]
\delta=
| ||||
| \sum | ||||
| n=0 |
| infty | |
| -\sum | |
| n=0 |
Cn+1\{e ⋅ n!\}-
| 1 | |
| 2 |
.
| infty | |
| \sum | |
| k=0 |
(-1)kk!=1-1+2-6+24-120+\ldots
| 1\delta | |
| = |
2-\cfrac{12}{4-\cfrac{22}{6-\cfrac{32}{8-\cfrac{42}{\ddots\cfrac{n2}{2(n+1)-...}}}}}
| 1\delta | |
| = |
1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{3}{1+\cfrac{3}{1+\cfrac{4}{...}}}}}}}
| 1 | |
| 1-\delta |
=3-\cfrac{2}{5-\cfrac{6}{7-\cfrac{12}{9-\cfrac{20}{\ddots\cfrac{n(n+1)}{2n+3-...}}}}}