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:
- $G = \ds \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u$
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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:
- $G = \cfrac 1 {2 - \cfrac {1^2} {4 - \cfrac {2^2} {6 - \cfrac {3^2} {8 - \cfrac {4^2} {10 - \dotsb} } } } }$
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Thus $G$ equals the integral form of the Euler-Gompertz constant.
$\blacksquare$
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,59634 7355 \ldots$