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Euler-Mascheroni Integrals

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Define

I_n=(-1)^nint_0^infty(lnz)^ne^(-z)dz,
(1)

then

I_n=(-1)^nGamma^((n))(1),
(2)

where Gamma^((n))(z) is the nth derivative of the gamma function.

Particular values include

I_0=1
(3)
I_1=gamma
(4)
I_2=gamma^2+1/6pi^2
(5)
I_3=gamma^3+1/2gammapi^2+2zeta(3)
(6)
I_4=gamma^4+gamma^2pi^2-3/(20)pi^4+8gammazeta(3),
(7)

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

See also

Euler-Mascheroni Constant, Gamma Function, Polygamma Function

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References

Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, pp. 213-214, 2004.Srivastava, H. M. and Choi, J. Series Associated with the Zeta and Related Functions. New York: Springer-verlag, 2006.

Referenced on Wolfram|Alpha

Euler-Mascheroni Integrals

Cite this as:

Weisstein, Eric W. "Euler-Mascheroni Integrals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MascheroniIntegrals.html

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