In mathematics, the **Gompertz constant** or **Euler-Gompertz constant**, denoted by
G
, appears in integral evaluations and as a value of special functions. It is named after B. Gompertz.

It can be defined by the continued fraction

G
=
1
2
−
1
4
−
4
6
−
9
8
−
16
10
−
25
12
−
36
14
−
49
16
−
…
,
or, alternatively, by

G
=
1
1
+
1
1
+
1
1
+
2
1
+
2
1
+
3
1
+
3
1
+
4
1
1
+
…
.
The most frequent appearance of
G
is in the following integrals:

G
=
∫
0
∞
ln
(
1
+
x
)
e
−
x
d
x
=
∫
0
∞
e
−
x
1
+
x
d
x
=
∫
0
1
1
1
−
log
(
x
)
d
x
.
The numerical value of
G
is about

G
=
0.596347362323194074341078499369279376074
…
During the studying divergent infinite series Euler met with
G
via, for example, the above integral representations. Le Lionnais called
G
as Gompertz constant by its role in survival analysis.

The constant
G
can be expressed by the exponential integral as

G
=
−
e
Ei
(
−
1
)
.
Applying the Taylor expansion of
Ei
we have that

G
=
−
e
(
γ
+
∑
n
=
1
∞
(
−
1
)
n
n
⋅
n
!
)
.
Gompertz's constant is connected to the Gregory coefficients via the 2013 formula of I. Mező:

G
=
∑
n
=
0
∞
ln
(
n
+
1
)
n
!
−
∑
n
=
0
∞
C
n
+
1
{
e
⋅
n
!
}
−
1
2
.