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Euler-Mascheroni Integrals

Define

\begin{displaymath}
I_n\equiv (-1)^n \int_0^\infty (\ln z)^n e^{-z}\,dz,
\end{displaymath} (1)

then
$\displaystyle I_0$ $\textstyle =$ $\displaystyle \int_0^\infty e^{-z}\,dz = [-e^{-z}]^\infty_0 = (0+1)=1$ (2)
$\displaystyle I_1$ $\textstyle =$ $\displaystyle -\int_0^\infty (\ln z)e^{-z}\, dz = \gamma$ (3)
$\displaystyle I_2$ $\textstyle =$ $\displaystyle \gamma^2+{\textstyle{1\over 6}}\pi^2$ (4)
$\displaystyle I_3$ $\textstyle =$ $\displaystyle \gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)$ (5)
$\displaystyle I_4$ $\textstyle =$ $\displaystyle \gamma^4+\gamma^2\pi^2-{\textstyle{3\over 20}}\pi^4+8\gamma\zeta(3),$ (6)

where $\gamma$ is the Euler-Mascheroni Constant and $\zeta(3)$ is Apéry's Constant.




© 1996-9 Eric W. Weisstein
1999-05-25