# Euler-Gompertz constant

The Euler-Gompertz constant is the value of the continued fraction^{}

$${C}_{2}=0+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{2}{1+\mathrm{\dots}}}}},$$ |

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:

$${C}_{2}=-e\text{Ei}(-1)={\int}_{1}^{\mathrm{\infty}}\frac{\text{exp}(1-x)}{x}\mathit{d}x,$$ |

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:

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

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

## References

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

Title | Euler-Gompertz constant |
---|---|

Canonical name | EulerGompertzConstant |

Date of creation | 2013-03-22 18:49:06 |

Last modified on | 2013-03-22 18:49:06 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 7 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A55 |

Synonym | Gompertz constant |