Hadjicostas's formula

In mathematics, Hadjicostas's formula is a formula relating a certain double integral to values of the Gamma function and the Riemann zeta function.

tatement

Let "s" be a complex number with Re("s") > −2. Then:$int\_0^1int\_0^1\; frac\{(-log(xy))^s\}\{1-xy\}(1-x),dx,dy=Gamma(s+2)left(zeta(s+2)-frac\{1\}\{s+1\}\; ight).$

Background

The first instance of the formula was proved and used by Frits Beukers in his 1978 paper giving an alternative proof of Apéry's theorem [F. Beukers, "A note on the irrationality of ζ(2) and ζ(3)", Bull. London Math. Soc. 11 (1979), pp. 268–272.] . He proved the formula when "s" = 0, and proved an equivalent formulation for the case "s" = 1. This led Petros Hadjicostas to conjecture the above formula in 2004 [P. Hadjicostas, [http://arxiv.org/abs/math.NT/0405423/ "A conjecture-generalization of Sondow’s formula"] , 2004, from the arXiv.] , and within a week it had been proven by Robin Chapman [R. Chapman, [http://arxiv.org/abs/math/0405478v2 "A proof of Hadjicostas’s conjecture"] , 2004, from the arXiv.] . He proved the formula holds when Re("s") > −1, and then extended the result by analytic continuation to get the full result.

pecial cases

As well as the two cases used by Beukers to get alternate expressions for ζ(2) and ζ(3), the formula can be used to express the Euler-Mascheroni constant as a double integral by letting "s" tend to −1:

:$gamma=-int\_0^1int\_0^1frac\{1-x\}\{(1-xy)log(xy)\},dx,dy.$

Notes