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In [[mathematics]], the '''Barnes G-function''' ''G''(''z'') is a [[function (mathematics)|function]] that is an extension of [[superfactorial]]s to the [[complex number]]s. It is related to the [[Gamma function]], the [[K-function]] and the [[Glaisher–Kinkelin constant]], and was named after [[mathematician]] [[Ernest William Barnes]].<ref>E.W.Barnes, "The theory of the G-function", ''Quarterly Journ. Pure and Appl. Math.'' '''31''' (1900), 264–314.</ref> Up to elementary factors, it is a special case of the [[double gamma function]].
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:<math> G(1+z)=(2\pi)^{z/2} \text{exp}\left(- \frac{z+z^2(1+\gamma)}{2} \right) \, \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)^k \text{exp}\left(\frac{z^2}{2k}-z\right) </math>
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where <math>\, \gamma \, </math> is the [[Euler–Mascheroni constant]], exp(''x'') = ''e''<sup>''x''</sup>, and is [[capital pi notation]].
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The Barnes ''G''-function satisfies the [[functional equation]]
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:<math> G(z+1)=\Gamma(z)\, G(z) </math>
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:<math>G(n)=\frac{(\Gamma(n))^{n-1}}{K(n)}</math>
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where <math>\,\Gamma(x)\,</math> denotes the [[Gamma function]] and ''K'' denotes the [[K-function]]. The functional equation uniquely defines the G function if the convexity condition: <math>\, \frac{d^3}{dx^3}G(x)\geq 0\, </math> is added.<ref>M. F. Vignéras, ''L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL<math>(2,\mathbb{Z})</math>'', Astérisque '''61''', 235–249 (1979).</ref>
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The [[difference equation]] for the G function, in conjunction with the [[functional equation]] for the [[Gamma function]], can be used to obtain the following [[reflection formula]] for the Barnes G function (originally proved by [[Hermann Kinkelin]]):
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:<math> \log G(1-z) = \log G(1+z)-z\log 2\pi+ \int_0^z \pi x \cot \pi x \, dx.</math>
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The logtangent integral on the right-hand side can be evaluated in terms of the [[Clausen function]] (of order 2), as is shown below:
 
 
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:<math>2\pi \log\left( \frac{G(1-z)}{G(1+z)} \right)= 2\pi z\log\left(\frac{\sin\pi z}{\pi} \right)+\text{Cl}_2(2\pi z)</math>
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The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation <math>\, Lc(z)\, </math> for the logtangent integral, and using the fact that <math>\,(d/dx) \log(\sin\pi x)=\pi\cot\pi x\,</math>, an integration by parts gives
 
 
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:<math>Lc(z)=\int_0^z\pi x\log(\sin \pi x)\,dx=z\log(\sin \pi z)-\int_0^z\log(\sin \pi x)\,dx=</math>
</ul>
 
 
:<math>z\log(\sin \pi z)-\int_0^z\Bigg[\log(2\sin \pi x)-\log 2\Bigg]\,dx=</math>
 
 
:<math>z\log(2\sin \pi z)-\int_0^z\log(2\sin \pi x)\,dx</math>
 
 
Performing the integral substitution <math>\, y=2\pi x \Rightarrow dx=dy/(2\pi)\,</math> gives
 
 
:<math>z\log(2\sin \pi z)-\frac{1}{2\pi}\int_0^{2\pi z}\log\left(2\sin \pi \frac{y}{2} \right)\,dy</math>
 
 
The [[Clausen function]] - of second order - has the integral representation
 
 
:<math>\text{Cl}_2(\theta) = -\int_0^{\theta}\log\Bigg|2\sin \frac{x}{2} \Bigg|\,dx</math>
 
 
However, within the interval <math>\, 0 < \theta < 2\pi \,</math>, the [[absolute value]] sign within the [[integrand]] can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the logtangent itegral, the following relation clearly holds:
 
 
:<math>Lc(z)=z\log(2\sin \pi z)+\frac{1}{2\pi}\, \text{Cl}_2(2\pi z)</math>
 
 
Thus, after a slight rearrangement of terms, the proof is complete:
 
 
:<math>2\pi \log\left( \frac{G(1-z)}{G(1+z)} \right)= 2\pi z\log\left(\frac{\sin\pi z}{\pi} \right)+\text{Cl}_2(2\pi z)\, . \, \Box </math>
 
 
Using the relation <math>\, G(1+z)=\Gamma(z)\, G(z) \,</math> and dividing the reflection formula by a factor of <math>\, 2\pi \,</math> gives the equivalent form:
 
 
:<math> \log\left( \frac{G(1-z)}{G(z)} \right)= z\log\left(\frac{\sin\pi z}{\pi}
\right)+\log\Gamma(z)+\frac{1}{2\pi}\text{Cl}_2(2\pi z) </math>
 
 
Ref: see '''Adamchik''' below for an equivalent form of the [[reflection formula]], but with a different proof.
 
==Reflection formula 2.0==
 
 
Replacing '''z''' with '''(1/2)-z''' in the previous reflection formula gives,  after some simplification, the equivalent formula shown below (involving [[Bernoulli polynomials]]):
 
 
:<math>\log\left( \frac{ G\left(\frac{1}{2}+z\right) }{ G\left(\frac{1}{2}-z\right) } \right) =</math>
 
 
:<math>
\log \Gamma \left(\frac{1}{2}-z \right) + B_1(z) \log 2\pi-\frac{1}{2}\log 2+\pi \int_0^z B_1(x) \tan \pi x \,dx</math>
 
 
 
==Taylor series expansion==
 
 
By [[Taylor's theorem]], and considering the logarithmic [[derivatives]] of the Barnes function, the following series expansion can be obtained:
 
 
:<math>\log G(1+z)= \frac{z}{2}\log 2\pi -\left( \frac{z+(1+\gamma)z^2}{2} \right) + \sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1}z^{k+1}</math>
 
 
It is valid for <math>\, 0 < z < 1 \,</math>. Here, <math>\, \zeta(x) \,</math> is the [[Riemann Zeta function]]:
 
 
:<math> \zeta(x)=\sum_{k=1}^{\infty}\frac{1}{k^x} </math>
 
 
Exponentiating both sides of the Taylor expansion gives:
 
 
:<math> G(1+z)=\exp \left[ \frac{z}{2}\log 2\pi -\left( \frac{z+(1+\gamma)z^2}{2} \right) + \sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1}z^{k+1} \right]=</math>
 
 
:<math>(2\pi)^{z/2}\text{exp}\left( -\frac{z+(1+\gamma)z^2}{2} \right) \exp \left[\sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1}z^{k+1} \right]</math>
 
 
Comparing this with the [[Weierstrass product]] form of the Barnes function gives the following relation:
 
 
:<math>\exp \left[\sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1}z^{k+1} \right] = \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)^k\text{exp}\left(\frac{z^2}{2k}-z\right)</math>
 
==Multiplication formula==
 
 
Like the Gamma function, the G-function also has a multiplication formula:<ref>I. Vardi, ''Determinants of Laplacians and multiple gamma functions'', SIAM J. Math. Anal. '''19''', 493–507 (1988).</ref>
 
 
:<math>
G(nz)= K(n) n^{n^{2}z^{2}/2-nz} (2\pi)^{-\frac{n^2-n}{2}z}\prod_{i=0}^{n-1}\prod_{j=0}^{n-1}G\left(z+\frac{i+j}{n}\right)
</math>
 
 
where <math>K(n)</math> is a constant given by:
 
 
:<math> K(n)= e^{-(n^2-1)\zeta^\prime(-1)} \cdot
n^{\frac{5}{12}}\cdot(2\pi)^{(n-1)/2}\,=\,
(Ae^{-\frac{1}{12}})^{n^2-1}\cdot n^{\frac{5}{12}}\cdot (2\pi)^{(n-1)/2}.</math>
 
 
Here <math>\zeta^\prime</math> is the derivative of the [[Riemann zeta function]] and <math>A</math> is the [[Glaisher–Kinkelin constant]].
 
== Asymptotic expansion ==
 
 
The [[logarithm]] of ''G''(''z'' + 1) has the following asymptotic expansion, as established by Barnes:
 
 
:<math> \log G(z+1)=</math>
 
 
:<math>\frac{1}{12}~-~\log A~+~\frac{z}{2}\log 2\pi~+~\left(\frac{z^2}{2} -\frac{1}{12}\right)\log z~-~\frac{3z^2}{4}~+~
\sum_{k=1}^{N}\frac{B_{2k + 2}}{4k\left(k + 1\right)z^{2k}}~+~O\left(\frac{1}{z^{2N + 2}}\right).</math>
 
 
Here the <math>B_{k}</math> are the [[Bernoulli numbers]] and <math>A</math> is the [[Glaisher–Kinkelin constant]]. (Note that somewhat confusingly at the time of Barnes <ref>E.T.Whittaker and G.N.Watson, "A course of modern analysis", CUP.</ref> the [[Bernoulli number]] <math>B_{2k}</math> would have been written as <math>(-1)^{k+1} B_k </math>, but this convention is no longer current.) This expansion is valid for <math>z </math> in any sector not containing the negative real axis with <math>|z|</math> large.
 
==Relation to the Loggamma integral==
 
 
The parametric Loggamma can be evaluated in terms of the Barnes G-function (Ref: this result is found in '''Adamchik''' below, but stated without proof):
 
 
:<math> \int_0^z \log \Gamma(x)\,dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi +z\log\Gamma(z) -\log G(1+z) </math>
 
 
The proof is somewhat indirect, and involves first considering the logarithmic difference of the [[Gamma function]] and Barnes G-function:
 
:<math>z\log \Gamma(z)-\log G(1+z)</math>
 
 
Where
 
 
:<math>\frac{1}{\Gamma(z)}= z e^{\gamma} \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}</math>
 
 
and <math>\,\gamma\,</math> is the [[Euler-Mascheroni constant]].
 
 
Taking the logarithm of the [[Weierstrass product]] forms of the Barnes function and Gamma function gives:
 
 
:<math>z\log \Gamma(z)-\log G(1+z)=-z \log\left(\frac{1}{\Gamma (z)}\right)-\log G(1+z)=</math>
 
 
:<math>-z \left[ \log z+\gamma z +\sum_{k=1}^{\infty} \Bigg\{ \log\left(1+\frac{z}{k} \right) -\frac{z}{k} \Bigg\} \right]</math>
 
 
:<math>-\left[ \frac{z}{2}\log 2\pi -\frac{z}{2}-\frac{z^2}{2} -\frac{z^2 \gamma}{2} + \sum_{k=1}^{\infty} \Bigg\{k\log\left(1+\frac{z}{k}\right) +\frac{z^2}{2k} -z \Bigg\} \right]</math>
 
 
A little simplification and re-ordering of terms gives the series expansion:
 
 
:<math> \sum_{k=1}^{\infty} \Bigg\{ (k+z)\log \left(1+\frac{z}{k}\right)-\frac{z^2}{2k}-z \Bigg\}=</math>
 
 
:<math>-z\log z-\frac{z}{2}\log 2\pi +\frac{z}{2} +\frac{z^2}{2}- \frac{z^2 \gamma}{2}- z\log\Gamma(z) +\log G(1+z)</math>
 
 
Finally, take the logarithm of the [[Weierstrass product]] form of the [[Gamma function]], and integrate over the interval <math>\, [0,\,z]\, </math> to obtain:
 
 
:<math>\int_0^z\log\Gamma(x)\,dx=-\int_0^z \log\left(\frac{1}{\Gamma(x)}\right)\,dx=</math>
 
 
:<math>-(z\log z-z)-\frac{z^2 \gamma}{2}- \sum_{k=1}^{\infty} \Bigg\{ (k+z)\log \left(1+\frac{z}{k}\right)-\frac{z^2}{2k}-z \Bigg\}</math>
 
 
Equating the two evaluations completes the proof:
 
 
:<math> \int_0^z \log \Gamma(x)\,dx=\frac{z(1-z)}{2}+\frac{z}{2}\log 2\pi +z\log\Gamma(z) -\log G(1+z)\, . \, \Box</math>
 
==References==
<references/>
 
*{{dlmf|first=R.A. |last=Askey|first2=R.|last2=Roy|id=5.17}}
{{DEFAULTSORT:Barnes G-Function}}
[[Category:Number theory]]
[[Category:Special functions]]
 
*{{cite web|last=Adamchik|first=Viktor S.|title=Contributions to the Theory of the Barnes function|url=http://arxiv.org/pdf/math/0308086v1.pdf|accessdate=2003}}

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What Esraa group allowed people to do was to join an ever more large set of individuals who wanted to make a statement about the political situation in Egypt individuals who didn need to be localized in any one place. As Hassan Khalil notes, people of Facebook are sons of the middle class throughout the country. After all, as strong as the Egyptian state might be, it cannot go around arresting 70,000 people, most of them wealthy and connected elites, especially if all they done is stay at home.

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Our two Presidents have reaffirmed their willingness to grow bilateral cooperation, including between intelligence agencies in the fight against international terrorism, the statement said, provocative Cold Warstyle actions don't contribute to building mutual trust. Was detained Monday night or early Tuesday morning, reportedly whilst in the act of trying to recruit the unnamed Russian. Based on Russian news agencies, he was detained on Akademika Pilyugina Street, in a residential neighborhood across from the park in southwest Moscow.

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The information you share, including what might otherwise be Protected Health Information, to this site is by design open to the public and is not a private, secure service. You need to think carefully before disclosing any personal information in any public forum. Just like any public forum on any web site, this information may also appear in thirdparty engines like google, MSN, Yahoo, etc.

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Since the attributes and traits of international manufacturers' representatives, visavis domestic representatives, happen to be somewhat scrambled, I want to point out how international agencies might come under a new classification. If under the current taxonomy, domestic manufacturers' representatives really are a "species," then international manufacturers' representative agencies, as a "subspecies," might best be named "International Information Brokers" (IIB). Besides selling the manufacturer's products or services to foreign buyers, what greatly distinguishes the IIB is: They have developed the skills for the remote retrieval of information and the organization of that information for manufacturers that need to sell internationally.

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Finding names Abercrombie Singapore for online retailers is only likely to become tougher because of the pace at which the market continues to be growing. The business is relatively new in India compared with the West, but several beginners have sprung up in the past couple of years while popular retailers for example Flipkart, Yebhi, Jabong and Myntra have expanded, mostly driven by demand from smaller towns and cities. Consultancy firm Technopak pegs the online retail market in India at $500 million and it has predicted it will grow to $70 billion by 2020..

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For retail operations, total revenue is used as the basis for judging whether a company is small. In general, the maximum size ranges from $7 million (hardware stores) to $30 million (boat stores and electronics shops). But there might be large variations even inside the same industry: A shop that sells women's clothes are considered "small" by the government whether it has revenue of under $25 million, but a jewelry boutique with the same amount of revenue cannot make use of the SBA's smallbusiness services (it must have sales of $14 million or less to qualify)..

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