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Hadjicostas's Formula

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Hadjicostas's formula is a generalization of the double integral over the unit square

gamma=int_0^1int_0^1(x-1)/((1-xy)ln(xy))dxdy
(1)

(Sondow 2003; Borwein et al. 2004, p. 49; Sondow 2005), where gamma is the Euler-Mascheroni constant. It states

int_0^1int_0^1(1-x)/(1-xy)[-ln(xy)]^sdxdy=Gamma(s+2)[zeta(s+2)-1/(s+1)]
(2)

for Re[s]>-2, where Gamma(z) is the gamma function and zeta(s) is the Riemann zeta function (although care must be taken at s=-1 because of the removable singularity present there). It was conjectured by Hadjicostas (2004) and almost immediately proved by Chapman (2004). The special case s=0 is related to Beukers's integral for zeta(2),

int_0^1int_0^1(dxdy)/(1-xy)=zeta(2)
(3)

(Beukers 1979). At s=1, the formula is related to Beukers's integral for Apéry's constant zeta(3), which is how interest in this class of integrals originally arose.

There is an analogous formula

int_0^1int_0^1(1-x)/(1+xy)[-ln(xy)]^sdxdy=Gamma(s+2)[eta(s+2)+(1-2eta(s+1))/(s+1)]
(4)

for Re[s]>-3, due to Sondow (2005), where eta(z) is the Dirichlet eta function. The apparent singularities at s=-1 and s=-2 are removable. This includes the special cases

ln(4/pi)=sum_(n=1)^(infty)(-1)^(n-1)[1/n-ln((n+1)/n)]
(5)
=int_0^1int_0^1(x-1)/((1+xy)ln(xy))dxdy
(6)
=0.241564...
(7)

(OEIS A094640; Sondow 2005) and

int_0^1int_0^1(1-x)/((1+xy)[ln(xy)]^2)dxdy=ln((pi^(1/2)A^6)/(2^(7/6)e))
(8)
=0.256220094...
(9)

(OEIS A103130), where A is the Glaisher-Kinkelin constant (Sondow 2005).

See also

Apéry's Constant, Euler-Mascheroni Constant, Riemann Zeta Function zeta(2), Unit Square Integral

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References

Beukers, F. "A Note on the Irrationality of zeta(2) and zeta(3)." Bull. London Math. Soc. 11, 268-272, 1979.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Chapman, R. "A Proof of Hadjicostas's Conjecture." 15 Jun 2004. https://arxiv.org/abs/math/0405478.Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants via Analytic Continuations of Lerch's Transcendent." Ramanujan J. 16, 247-270, 2008. https://doi.org/10.1007/s11139-007-9102-0.Hadjicostas, P. "A Conjecture-Generalization of Sondow's Formula." 21 May 2004. https://arxiv.org/abs/math/0405423.Sloane, N. J. A. Sequences A094640 and A103130 in "The On-Line Encyclopedia of Integer Sequences."Sondow, J. "Criteria for Irrationality of Euler's Constant." Proc. Amer. Math. Soc. 131, 3335-3344, 2003. https://arxiv.org/abs/math/0209070.Sondow, J. "Double Integrals for Euler's Constant and ln(4/pi) and an Analog of Hadjicostas's Formula." Amer. Math. Monthly 112, 61-65, 2005.

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Hadjicostas's Formula

Cite this as:

Weisstein, Eric W. "Hadjicostas's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HadjicostassFormula.html

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