Euler-Mascheroni Integrals

Define

(1)

then

(2)

where  is the th derivative of the gamma function.

Particular values include

			(3)
			(4)
			(5)
			(6)
			(7)

where  is the Euler-Mascheroni constant and  is Apéry's constant.

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