Neha Patil (Editor)

Gompertz constant

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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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