integral representations of the Mascheroni constant

Mascheroni’s constant can be expressed by the following integrals:

$\gamma$ $=$ $-{\displaystyle {\int }_0^1}\log (-\log x)𝑑x$

$\gamma$ $=$ $-{\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-x}\log xdx$

$\gamma$ $=$ ${\displaystyle {\int }_0^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{e^t-1}}-{\displaystyle \frac{1}{t{e}^t}}\right)𝑑t$

$\gamma$ $=$ ${\displaystyle {\int }_0^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{t}}-{\displaystyle \frac{1}{1+t}}-{\displaystyle \frac{1}{t{e}^t}}\right)𝑑t$

Title	integral representations of the Mascheroni constant
Canonical name	IntegralRepresentationsOfTheMascheroniConstant
Date of creation	2013-03-22 15:53:24
Last modified on	2013-03-22 15:53:24
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	8
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 40A25