Definition:Euler-Gompertz Constant

Definition

Integral Form

The Euler-Gompertz constant is the real number $G$ defined as:

$G = \ds \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u$

As a Continued Fraction

The Euler-Gompertz constant is the real number $G$ defined as:

$G = \cfrac 1 {2 - \cfrac {1^2} {4 - \cfrac {2^2} {6 - \cfrac {3^2} {8 - \cfrac {4^2} {10 - \dotsb} } } } }$

Decimal Expansion

The decimal expansion of the Euler-Gompertz constant $G$ starts:

$G \approx 0 \cdotp 59634 \, 73623 \, 23194 \, 07434 \, 10784 \, 99369 \, 27937 \, 6074 \ldots$

Also known as

This constant is otherwise known as the Gompertz constant.

Also see

• Equivalence of Definitions of Euler-Gompertz Constant

Source of Name

This entry was named for Leonhard Paul Euler and Benjamin Gompertz.

Historical Note

The Euler-Gompertz constant was associated by Benjamin Gompertz in $1925$ with the rate of survival with respect to age.

Leonhard Paul Euler had previously encountered it during his study of divergent infinite series.

Thomas Joannes Stieltjes demonstrated the equivalence of the integral form and the continued fraction form.