Equivalence of Definitions of Euler-Gompertz Constant

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Theorem

The following definitions of the concept of Euler-Gompertz Constant are equivalent:

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:

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


Proof

Integral Form implies Continued Fraction Form

Let $G$ be the integral form of the Euler-Gompertz constant.

Then by definition:

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



Thus $G$ equals the continued fraction form of the Euler-Gompertz constant.

$\Box$


Continued Fraction Form implies Integral Form

Let $G$ be the continued fraction form of the Euler-Gompertz constant.

Then by definition:

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



Thus $G$ equals the integral form of the Euler-Gompertz constant.

$\blacksquare$


Sources