Gompertz constant - Alchetron, The Free Social Encyclopedia

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.

Identities involving the Gompertz constant

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 .

References

Gompertz constant Wikipedia

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