Hadjicostas's formula
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In mathematics, Hadjicostas's formula is a formula relating a certain double integral to values of the Gamma function and the Riemann zeta function.
Contents
1 Statement
2 Background
3 Special cases
4 Notes
5 See also
Statement
Let s be a complex number with s ≠ -1 and Re(s) > −2. Then
- ∫01∫011−x1−xy(−log(xy))sdxdy=Γ(s+2)(ζ(s+2)−1s+1).displaystyle int _0^1int _0^1frac 1-x1-xy(-log(xy))^s,dx,dy=Gamma (s+2)left(zeta (s+2)-frac 1s+1right).
Here Γ is the Gamma function and ζ is the Riemann zeta function.
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.[1] 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,[2] and within a week it had been proven by Robin Chapman.[3] He proved the formula holds when Re(s) > −1, and then extended the result by analytic continuation to get the full result.
Special 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:
- γ=∫01∫011−x(1−xy)(−log(xy))dxdy.displaystyle gamma =int _0^1int _0^1frac 1-x(1-xy)(-log(xy)),dx,dy.
The latter formula was first discovered by Jonathan Sondow[4] and is the one referred to in the title of Hadjicostas's paper.
Notes
^ Beukers, F. (1979). "A note on the irrationality of ζ(2) and ζ(3)". Bull. London Math. Soc. 11 (3): 268–272. doi:10.1112/blms/11.3.268..mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
^ Hadjicostas, P. (2004). "A conjecture-generalization of Sondow's formula". arXiv:math.NT/0405423.
^ Chapman, R. (2004). "A proof of Hadjicostas's conjecture". arXiv:math/0405478.
^ Sondow, J. (2003). "Criteria for irrationality of Euler's constant". Proc. Amer. Math. Soc. 131: 3335–3344. doi:10.1090/S0002-9939-03-07081-3.
See also
Hessami Pilehrood, Kh.; Hessami Pilehrood, T. (2008). "Vacca-type series for values of the generalized-Euler-constant function and its derivative". arXiv:0808.0410.
Sondow, J (2005). "Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula". American Mathematical Monthly. 112: 61–65. arXiv:math.CA/0211148. doi:10.2307/30037385.
Sondow, Jonathan; Hadjicostas, Petros (2008). "The generalized-Euler-constant function γ(z) and a generalization of Somos's quadratic recurrence constant". Journal of Mathematical Analysis and Applications. 332: 292–314. arXiv:math/0610499. doi:10.1016/j.jmaa.2006.09.081.