In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by
δ
{\displaystyle \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:
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 }
is about
δ = 0.596347362323194074341078499369... (sequence A073003 in the OEIS).
Contents
History
When Euler studied divergent infinite series, he encountered
δ
{\displaystyle \delta }
via, for example, the above integral representation. Le Lionnais called
δ
{\displaystyle \delta }
the Gompertz constant because of its role in survival analysis.
In 1962, Shidlovski 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, who proved that at least one of them is transcendental.
Identities involving the Gompertz constant
The most frequent appearance of
δ
{\displaystyle \delta }
is in the following integrals:
δ
=
∫
0
∞
ln
(
1
+
x
)
e
−
x
d
x
=
∫
0
1
1
1
−
ln
(
x
)
d
x
{\displaystyle \delta =\int _{0}^{\infty }\ln(1+x)e^{-x}dx=\int _{0}^{1}{\frac {1}{1-\ln(x)}}dx}