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# Euler-Gompertz constant

The Euler-Gompertz constant is the value of the continued fraction

$C_{2}=0+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{2}{1+\ldots}}}},$ |

in which after three appearances of 1 in the numerator position, follow the integers from 2 forward each given twice consecutively; the value of this constant is approximately 0.596347362323194074341078499369279376074… Finch gives two formulas for this constant:

$C_{2}=-e\textrm{Ei}(-1)=\int_{1}^{\infty}\frac{\textrm{exp}(1-x)}{x}dx,$ |

with $e$ being the natural log base and Ei being the exponential integral.

The constant can also be expressed as a formula involving an infinite sum:

$e\left(\left(\sum_{{i=1}}^{\infty}\frac{(-1)^{{i-1}}}{i!i}\right)-\gamma\right),$ |

with $\gamma$ being the Euler-Mascheroni constant.

# References

- 1 Steven R. Finch, Mathematical Constants. Cambridge: Cambridge University Press (2003): 424

Synonym:

Gompertz constant

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11A55*no label found*

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