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Typo/general fixing, replaced: assement → assessment using AWB

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	~~In [[mathematics]]~~, ~~'''Hadjicostas's formula'''~~ is ~~a formula relating a certain [[Multiple integral\|double integral]]~~ to ~~values of the [[Gamma function]]~~ and ~~the [[Riemann zeta function]].~~		Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. The effect is to promote older posts by moving them back onto the front page and into the rss feed. Donor oocytes and menopausal pregnancy: Oocyte donation to women of advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website. <br><br>Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. Wordpress have every reason with it which promote wordpress development. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Word - Press makes it possible to successfully and manage your website. <br><br>If you adored this short article and you would such as to obtain more information regarding [http://htxurl.info/wordpressbackup700794 wordpress backup plugin] kindly check out our own page. Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. After sending these details, your Word - Press blog will be setup within a few days. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Word - Press is the most popular open source content management system (CMS) in the world today. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. Automated deal feed integration option to populate your blog with relevant deals. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Page speed is an important factor in ranking, especially with Google. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order.

	~~==Statement==~~

	~~Let ''s'' be~~ a ~~[[complex number]]~~ with ~~Re(''s'') > −2~~. ~~Then~~
	~~:<math>\int_0^1\int_0^1 \frac{1-x}{1-xy}(-\log(xy))^s\,dx\,dy=\Gamma(s+2)\left(\zeta(s+2)-\frac{1}{s+1}\right).</math>~~

	~~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]]~~.<~~ref~~>~~{{cite journal\|first1=F.\|last1=Beukers\|title=A note on~~ the ~~irrationality of ζ(2) and ζ(3)\|journal=Bull. London Math. Soc.\|volume=11\|issue=3\|year=1979 \|pages=268–272\|doi=~~10.~~1112/blms/11~~.~~3.268}}</ref> 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~~,~~<ref>{{cite arxiv\|first1=P~~.~~\|last1=Hadjicostas\|eprint=math~~.~~NT/0405423\|title=A conjecture-generalization of Sondow’s formula\|year=2004}}~~<~~/ref~~> ~~and within a week it had been proven by Robin Chapman.~~<~~ref~~>~~{{cite arxiv\|first1=R~~.~~\|last1=Chapman\|eprint=math~~/~~0405478\|title=A proof~~ of ~~Hadjicostas’s conjecture\|year=2004}}</ref> 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:~~

	~~:<math>\gamma=\int_0^1\int_0^1\frac{1-x}{(1-xy)(-\log(xy))}\,dx\,dy~~.<~~/math~~>

	~~The latter formula was first discovered by Jonathan Sondow~~<~~ref~~>~~{{cite journal\|first1=J~~.~~\|last1=Sondow\|title=Criteria~~ for ~~irrationality of Euler's constant\|journal=Proc~~. ~~Amer. Math. Soc.\|volume=131\|year=2003\|pages= 3335–3344\|doi=10~~.~~1090/S0002~~-~~9939-03-07081-3 }}</ref> and~~ is the ~~one referred to~~ in the ~~title of Hadjicostas's paper~~.

	~~==Notes==~~
	~~{{reflist}}~~

	~~==See also==~~
	*{{cite arxiv
	~~\|first1=Kh.~~
	~~\|last1=Hessami Pilehrood~~
	~~\|first2=T.~~
	~~\|last2=Hessami Pilehrood~~
	~~\|title=Vacca-type series for values~~ of ~~the generalized-Euler-constant function~~ and ~~its derivative~~
	~~\|eprint=0808~~.~~0410~~
	~~\|year=2008~~
	}}

	*Sondow, J. ~~(2005). "[http://arXiv~~.~~org/abs/math.CA/0211148 Double integrals for Euler~~'s ~~constant and ln 4/π and~~ an ~~analog of Hadjicostas's formula~~,~~]" ''[[American Mathematical Monthly]]'' '''112''': 61-65~~.

	~~{{DEFAULTSORT:Hadjicostas's Formula}}~~
	~~[[Category:Zeta~~ and ~~L-functions]]~~

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

==Statement==

Let ''s'' be a [[complex number]] with Re(''s'') > −2. Then
:<math>\int_0^1\int_0^1 \frac{1-x}{1-xy}(-\log(xy))^s\,dx\,dy=\Gamma(s+2)\left(\zeta(s+2)-\frac{1}{s+1}\right).</math>

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]].<ref>{{cite journal|first1=F.|last1=Beukers|title=A note on the irrationality of ζ(2) and ζ(3)|journal=Bull. London Math. Soc.|volume=11|issue=3|year=1979 |pages=268–272|doi=10.1112/blms/11.3.268}}</ref> 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,<ref>{{cite arxiv|first1=P.|last1=Hadjicostas|eprint=math.NT/0405423|title=A conjecture-generalization of Sondow’s formula|year=2004}}</ref> and within a week it had been proven by Robin Chapman.<ref>{{cite arxiv|first1=R.|last1=Chapman|eprint=math/0405478|title=A proof of Hadjicostas’s conjecture|year=2004}}</ref> 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:

:<math>\gamma=\int_0^1\int_0^1\frac{1-x}{(1-xy)(-\log(xy))}\,dx\,dy.</math>

The latter formula was first discovered by Jonathan Sondow<ref>{{cite journal|first1=J.|last1=Sondow|title=Criteria for irrationality of Euler's constant|journal=Proc. Amer. Math. Soc.|volume=131|year=2003|pages= 3335–3344|doi=10.1090/S0002-9939-03-07081-3 }}</ref> and is the one referred to in the title of Hadjicostas's paper.

==Notes==
{{reflist}}

==See also==
*{{cite arxiv
|first1=Kh.
|last1=Hessami Pilehrood
|first2=T.
|last2=Hessami Pilehrood
|title=Vacca-type series for values of the generalized-Euler-constant function and its derivative
|eprint=0808.0410
|year=2008
}}

*Sondow, J. (2005). "[http://arXiv.org/abs/math.CA/0211148 Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula,]" ''[[American Mathematical Monthly]]'' '''112''': 61-65.

{{DEFAULTSORT:Hadjicostas's Formula}}
[[Category:Zeta and L-functions]]

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en>John of Reading: Typo/general fixing, replaced: assement → assessment using AWB

en>Helpful Pixie Bot: ISBNs (Build KE)