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Barnes G-function

Table of contents: Definitions - Illustrations - Domain - Logarithmic form - Specific values - Singularities and zeros - Functional equations - Derivatives and differential equations - Representation by other functions - Series and product representations - Integral representations - Bounds and inequalities - Matrix formulas

Definitions

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Symbol: BarnesG G(z)G(z) Barnes G-function
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Symbol: LogBarnesG logG(z)\log G(z) Logarithmic Barnes G-function

Illustrations

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Image: Plot of G(x)G(x) on x[4,6]x \in \left[-4, 6\right]
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Image: X-ray of G(z)G(z) on z[4,6]+[5,5]iz \in \left[-4, 6\right] + \left[-5, 5\right] i
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Image: X-ray of logG(z)\log G(z) on z[4,6]+[5,5]iz \in \left[-4, 6\right] + \left[-5, 5\right] i

Domain

Barnes G-function

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G(z) is holomorphic on zCG(z) \text{ is holomorphic on } z \in \mathbb{C}
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nZ        G(n)Z0n \in \mathbb{Z} \;\implies\; G(n) \in \mathbb{Z}_{\ge 0}
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xR        G(x)Rx \in \mathbb{R} \;\implies\; G(x) \in \mathbb{R}
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zC        G(z)Cz \in \mathbb{C} \;\implies\; G(z) \in \mathbb{C}

Logarithmic Barnes G-function

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logG(z) is holomorphic on zC(,0]\log G(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]
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x(0,)        logG(x)Rx \in \left(0, \infty\right) \;\implies\; \log G(x) \in \mathbb{R}
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zC{0,1,}        logG(z)Cz \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \log G(z) \in \mathbb{C}
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z{0,1,}        logG(z){}z \in \{0, -1, \ldots\} \;\implies\; \log G(z) \in \left\{-\infty\right\}

Logarithmic form

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G(z)=exp ⁣(logG(z))G(z) = \exp\!\left(\log G(z)\right)
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logG(x)={log ⁣(G(x)),x>0log ⁣(G(x))+12n(n1)πi,otherwise   where n=x\log G(x) = \begin{cases} \log\!\left(G(x)\right), & x > 0\\\log\!\left(\left|G(x)\right|\right) + \frac{1}{2} n \left(n - 1\right) \pi i, & \text{otherwise}\\ \end{cases}\; \text{ where } n = \left\lfloor x \right\rfloor
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(z(0,)orz2.5<2.5)    (logG(z)=log ⁣(G(z)))\left(z \in \left(0, \infty\right) \,\mathbin{\operatorname{or}}\, \left|z - 2.5\right| < 2.5\right) \implies \left(\log G(z) = \log\!\left(G(z)\right)\right)

Specific values

Integers

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G(n)={k=1n2k!,n10,n0G(n) = \begin{cases} \prod_{k=1}^{n - 2} k !, & n \ge 1\\0, & n \le 0\\ \end{cases}
daef08
logG(n)={log ⁣(G(n)),n1,n0\log G(n) = \begin{cases} \log\!\left(G(n)\right), & n \ge 1\\-\infty, & n \le 0\\ \end{cases}
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Table of G(n)G(n) for 0n150 \le n \le 15
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Table of G ⁣(10n)G\!\left({10}^{n}\right) to 50 digits for 0n100 \le n \le 10

Rational arguments

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G ⁣(12)=21/24e1/8π1/4A3/2G\!\left(\frac{1}{2}\right) = \frac{{2}^{1 / 24} {e}^{1 / 8}}{{\pi}^{1 / 4} {A}^{3 / 2}}
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G ⁣(14)=e3/32G/4πA9/8(Γ ⁣(14))3/4G\!\left(\frac{1}{4}\right) = \frac{{e}^{3 / 32 - G / 4 \pi}}{{A}^{9 / 8} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{3 / 4}}
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G ⁣(34)=e3/32+G/4π(Γ ⁣(14))1/421/8π1/4A9/8G\!\left(\frac{3}{4}\right) = \frac{{e}^{3 / 32 + G / 4 \pi} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{1 / 4}}{{2}^{1 / 8} {\pi}^{1 / 4} {A}^{9 / 8}}

Derivatives

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G(0)=1G'(0) = 1
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G(1)=log ⁣(2π)12G'(1) = \frac{\log\!\left(2 \pi\right) - 1}{2}
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G(2)=log ⁣(2π)12γG'(2) = \frac{\log\!\left(2 \pi\right) - 1}{2} - \gamma
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G(n)={0,n<01,n=012(log ⁣(2π)1),n=1G(n)(12log ⁣(2π)+(n1)(Hn2γ1)+12),n2G'(n) = \begin{cases} 0, & n < 0\\1, & n = 0\\\frac{1}{2} \left(\log\!\left(2 \pi\right) - 1\right), & n = 1\\G(n) \left(\frac{1}{2} \log\!\left(2 \pi\right) + \left(n - 1\right) \left(H_{n - 2} - \gamma - 1\right) + \frac{1}{2}\right), & n \ge 2\\ \end{cases}

Singularities and zeros

Zeros

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zeroszCG(z)={0,1,}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} G(z) = \{0, -1, \ldots\}
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ordz=nG(z)=n+1\mathop{\operatorname{ord}}\limits_{z=-n} G(z) = n + 1
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zeroszClogG(z)={1,2,3}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log G(z) = \left\{1, 2, 3\right\}

Branch cuts

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BranchPoints ⁣(logG(z),z,C)={0,1,}\operatorname{BranchPoints}\!\left(\log G(z), z, \mathbb{C}\right) = \{0, -1, \ldots\}
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BranchCuts ⁣(logG(z),z,C)={(n1,n):nZ0}\operatorname{BranchCuts}\!\left(\log G(z), z, \mathbb{C}\right) = \left\{ \left(-n - 1, -n\right) : n \in \mathbb{Z}_{\ge 0} \right\}
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Im ⁣(logG(x))=n(n1)2π   where n=x\operatorname{Im}\!\left(\log G(x)\right) = \frac{n \left(n - 1\right)}{2} \pi\; \text{ where } n = \left\lfloor x \right\rfloor
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limε0+[logG ⁣(x+εi)]=logG(x)\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x + \varepsilon i\right)\right] = \log G(x)
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limε0+[logG ⁣(xεi)]=logG(x)=logG(x)n(n1)πi   where n=x\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x - \varepsilon i\right)\right] = \overline{\log G(x)} = \log G(x) - n \left(n - 1\right) \pi i\; \text{ where } n = \left\lfloor x \right\rfloor

Functional equations

Recurrence relation

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G ⁣(z+1)=Γ(z)G(z)G\!\left(z + 1\right) = \Gamma(z) G(z)
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logG ⁣(z+1)=logΓ(z)+logG(z)\log G\!\left(z + 1\right) = \log \Gamma(z) + \log G(z)
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G ⁣(z+n)=[k=1n(z+k1)nk](Γ(z))nG(z)G\!\left(z + n\right) = \left[\prod_{k=1}^{n} {\left(z + k - 1\right)}^{n - k}\right] {\left(\Gamma(z)\right)}^{n} G(z)

Reflection formula, real variables

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G ⁣(1x)=(1)(x1)/2+1G ⁣(1+x)(sin ⁣(πx)π)xexp ⁣(12πIm ⁣(Li2 ⁣(e2πix)))G\!\left(1 - x\right) = {\left(-1\right)}^{\left\lfloor \left( x - 1 \right) / 2 \right\rfloor + 1} G\!\left(1 + x\right) {\left(\frac{\left|\sin\!\left(\pi x\right)\right|}{\pi}\right)}^{x} \exp\!\left(\frac{1}{2 \pi} \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left({e}^{2 \pi i x}\right)\right)\right)
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logG ⁣(1x)=logG ⁣(1+x)+log ⁣(sin ⁣(πx)π)12πIm ⁣(Li2 ⁣(e2πix))+sgn(x)n(n+1)πi2   where n=x\log G\!\left(1 - x\right) = \log G\!\left(1 + x\right) + \log\!\left(\frac{\left|\sin\!\left(\pi x\right)\right|}{\pi}\right) \frac{1}{2 \pi} \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left({e}^{2 \pi i x}\right)\right) + \operatorname{sgn}(x) n \left(n + 1\right) \frac{\pi i}{2}\; \text{ where } n = \left\lfloor x \right\rfloor

Reflection formula, complex variables

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logG ⁣(1z)=logG ⁣(1+z)log ⁣(2π)z+0zπxcot ⁣(πx)dx\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) - \log\!\left(2 \pi\right) z + \int_{0}^{z} \pi x \cot\!\left(\pi x\right) \, dx
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logG ⁣(1z)=logG ⁣(1+z)log ⁣(2π)z+{0iπxcot ⁣(πx)dx+izπxcot ⁣(πx)dx,1<Re(z)<1orIm(z)>0or(Im(z)=0andRe(z)<1)0iπxcot ⁣(πx)dx+izπxcot ⁣(πx)dx,1<Re(z)<1orIm(z)<0or(Im(z)=0andRe(z)>1)\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) - \log\!\left(2 \pi\right) z + \begin{cases} \int_{0}^{i} \pi x \cot\!\left(\pi x\right) \, dx + \int_{i}^{z} \pi x \cot\!\left(\pi x\right) \, dx, & -1 < \operatorname{Re}(z) < 1 \,\mathbin{\operatorname{or}}\, \operatorname{Im}(z) > 0 \,\mathbin{\operatorname{or}}\, \left(\operatorname{Im}(z) = 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) < 1\right)\\\int_{0}^{-i} \pi x \cot\!\left(\pi x\right) \, dx + \int_{-i}^{z} \pi x \cot\!\left(\pi x\right) \, dx, & -1 < \operatorname{Re}(z) < 1 \,\mathbin{\operatorname{or}}\, \operatorname{Im}(z) < 0 \,\mathbin{\operatorname{or}}\, \left(\operatorname{Im}(z) = 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) > -1\right)\\ \end{cases}
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logG ⁣(1z)=logG ⁣(1+z)+{F(z),0<Re(z)<1orIm(z)>0or(Im(z)=0andRe(z)<1)F ⁣(z),1<Re(z)<0orIm(z)<0or(Im(z)=0andRe(z)>1)   where F(z)=πi2(z2z+16)z(logΓ(z)+logΓ ⁣(1z))i2πLi2 ⁣(e2πiz)\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) + \begin{cases} F(z), & 0 < \operatorname{Re}(z) < 1 \,\mathbin{\operatorname{or}}\, \operatorname{Im}(z) > 0 \,\mathbin{\operatorname{or}}\, \left(\operatorname{Im}(z) = 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) < 1\right)\\-F\!\left(-z\right), & -1 < \operatorname{Re}(z) < 0 \,\mathbin{\operatorname{or}}\, \operatorname{Im}(z) < 0 \,\mathbin{\operatorname{or}}\, \left(\operatorname{Im}(z) = 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) > -1\right)\\ \end{cases}\; \text{ where } F(z) = \frac{\pi i}{2} \left({z}^{2} - z + \frac{1}{6}\right) - z \left(\log \Gamma(z) + \log \Gamma\!\left(1 - z\right)\right) - \frac{i}{2 \pi} \operatorname{Li}_{2}\!\left({e}^{2 \pi i z}\right)

Multiplication theorem

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G ⁣(nz)=e(log(A)1/12)(n21)nn2z2/2nz+5/12(2π)(n1)(1nz)/2i=0n1j=0n1G ⁣(z+i+jn)G\!\left(n z\right) = {e}^{\left(\log(A) - 1 / 12\right) \left({n}^{2} - 1\right)} {n}^{{n}^{2} {z}^{2} / 2 - n z + 5 / 12} {\left(2 \pi\right)}^{\left(n - 1\right) \left(1 - n z\right) / 2} \prod_{i=0}^{n - 1} \prod_{j=0}^{n - 1} G\!\left(z + \frac{i + j}{n}\right)

Conjugate symmetry

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G ⁣(z)=G(z)G\!\left(\overline{z}\right) = \overline{G(z)}
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logG ⁣(z)={logG(z),z(,0]logG(z),otherwise\log G\!\left(\overline{z}\right) = \begin{cases} \log G(z), & z \in \left(-\infty, 0\right]\\\overline{\log G(z)}, & \text{otherwise}\\ \end{cases}

Derivatives and differential equations

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G(z)=G(z)((z1)ψ ⁣(z)z+log ⁣(2π)+12)G'(z) = G(z) \left(\left(z - 1\right) \psi\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right) + 1}{2}\right)
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ddz[logG(z)]=(z1)ψ ⁣(z)z+log ⁣(2π)+12\frac{d}{d z}\, \left[\log G(z)\right] = \left(z - 1\right) \psi\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right) + 1}{2}

Representation by other functions

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logG(z)=(z1)logΓ(z)[ddsζ ⁣(s,z)]s=1+ζ(1)\log G(z) = \left(z - 1\right) \log \Gamma(z) - \left[ \frac{d}{d s}\, \zeta\!\left(s, z\right) \right]_{s = -1} + \zeta'(-1)

Series and product representations

Taylor series

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logG ⁣(1+z)=log ⁣(2π)12z1+γ2z2+n=3(1)n+1ζ ⁣(n1)nzn\log G\!\left(1 + z\right) = \frac{\log\!\left(2 \pi\right) - 1}{2} z - \frac{1 + \gamma}{2} {z}^{2} + \sum_{n=3}^{\infty} \frac{{\left(-1\right)}^{n + 1} \zeta\!\left(n - 1\right)}{n} {z}^{n}

Weierstrass product

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G ⁣(z+1)=(2π)z/2e(z+(γ+1)z2)/2k=1[(1+zk)kexp ⁣(z22kz)]G\!\left(z + 1\right) = {\left(2 \pi\right)}^{z / 2} {e}^{-\left( z + \left(\gamma + 1\right) {z}^{2} \right) / 2} \prod_{k=1}^{\infty} \left[{\left(1 + \frac{z}{k}\right)}^{k} \exp\!\left(\frac{{z}^{2}}{2 k} - z\right)\right]

Asymptotic expansion

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Symbol: LogBarnesGRemainder RN ⁣(z)R_{N}\!\left(z\right) Remainder term in asymptotic expansion of logarithmic Barnes G-function
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logG ⁣(z+1)=z24+zlogΓ ⁣(z+1)(z(z+1)2+112)log(z)log(A)+n=1N1B2n+22n(2n+1)(2n+2)z2n+RN ⁣(z)\log G\!\left(z + 1\right) = \frac{{z}^{2}}{4} + z \log \Gamma\!\left(z + 1\right) - \left(\frac{z \left(z + 1\right)}{2} + \frac{1}{12}\right) \log(z) - \log(A) + \sum_{n=1}^{N - 1} \frac{B_{2 n + 2}}{2 n \left(2 n + 1\right) \left(2 n + 2\right) {z}^{2 n}} + R_{N}\!\left(z\right)
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RN ⁣(z)=0(t2coth ⁣(t2)B2k(2k)!t2k)eztt3dtR_{N}\!\left(z\right) = \int_{0}^{\infty} \left(\frac{t}{2} \coth\!\left(\frac{t}{2}\right) - \sum \frac{B_{2 k}}{\left(2 k\right)!} {t}^{2 k}\right) \frac{{e}^{-z t}}{{t}^{3}} \, dt
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RN ⁣(z)=1z2N(1)N+1π0t2N11+(tz)2Li2 ⁣(e2πt)dtR_{N}\!\left(z\right) = \frac{1}{{z}^{2 N}} \frac{{\left(-1\right)}^{N + 1}}{\pi} \int_{0}^{\infty} \frac{{t}^{2 N - 1}}{1 + {\left(\frac{t}{z}\right)}^{2}} \operatorname{Li}_{2}\!\left({e}^{-2 \pi t}\right) \, dt
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RN ⁣(z)=12N(2N+1)0B2N+1 ⁣(tt)(t+z)2NdtR_{N}\!\left(z\right) = \frac{1}{2 N \left(2 N + 1\right)} \int_{0}^{\infty} \frac{B_{2 N + 1}\!\left(t - \left\lfloor t \right\rfloor\right)}{{\left(t + z\right)}^{2 N}} \, dt
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RN ⁣(z)B2N+22N(2N+1)(2N+2)z2N{1,arg(z)π4sec2N+1 ⁣(12arg(z)),arg(z)<π\left|R_{N}\!\left(z\right)\right| \le \frac{\left|B_{2 N + 2}\right|}{2 N \left(2 N + 1\right) \left(2 N + 2\right) {\left|z\right|}^{2 N}} \begin{cases} 1, & \left|\arg(z)\right| \le \frac{\pi}{4}\\\sec^{2 N + 1}\!\left(\frac{1}{2} \arg(z)\right), & \left|\arg(z)\right| < \pi\\ \end{cases}

Integral representations

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logG ⁣(z+1)=z(1z)2+z2log ⁣(2π)+zlogΓ(z)0zlogΓ(x)dx\log G\!\left(z + 1\right) = \frac{z \left(1 - z\right)}{2} + \frac{z}{2} \log\!\left(2 \pi\right) + z \log \Gamma(z) - \int_{0}^{z} \log \Gamma(x) \, dx
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logG ⁣(z+1)=z(1z)2+z2log ⁣(2π)+0zxψ ⁣(x)dx\log G\!\left(z + 1\right) = \frac{z \left(1 - z\right)}{2} + \frac{z}{2} \log\!\left(2 \pi\right) + \int_{0}^{z} x \psi\!\left(x\right) \, dx
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logG ⁣(z+1)=z24(2log(z)3)+zlog ⁣(2π)2+112log(A)0xlog ⁣(x2+z2)e2πx1dx\log G\!\left(z + 1\right) = \frac{{z}^{2}}{4} \left(2 \log(z) - 3\right) + \frac{z \log\!\left(2 \pi\right)}{2} + \frac{1}{12} - \log(A) - \int_{0}^{\infty} \frac{x \log\!\left({x}^{2} + {z}^{2}\right)}{{e}^{2 \pi x} - 1} \, dx
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logG ⁣(z+1)=zlogΓ(z)+z24log(z)2B2 ⁣(z)log(A)0ezxx2(11ex1x12x12)dx\log G\!\left(z + 1\right) = z \log \Gamma(z) + \frac{{z}^{2}}{4} - \frac{\log(z)}{2} B_{2}\!\left(z\right) - \log(A) - \int_{0}^{\infty} \frac{{e}^{-z x}}{{x}^{2}} \left(\frac{1}{1 - {e}^{-x}} - \frac{1}{x} - \frac{1}{2} - \frac{x}{12}\right) \, dx

Bounds and inequalities

Upper and lower bounds

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logG ⁣(x+1)<(x22112)log(x)3x24+log ⁣(2π)2x+112log(A)\log G\!\left(x + 1\right) < \left(\frac{{x}^{2}}{2} - \frac{1}{12}\right) \log(x) - \frac{3 {x}^{2}}{4} + \frac{\log\!\left(2 \pi\right)}{2} x + \frac{1}{12} - \log(A)
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logG ⁣(x+1)>(x22112)log(x)3x24+log ⁣(2π)2x+112log(A)1240x2\log G\!\left(x + 1\right) > \left(\frac{{x}^{2}}{2} - \frac{1}{12}\right) \log(x) - \frac{3 {x}^{2}}{4} + \frac{\log\!\left(2 \pi\right)}{2} x + \frac{1}{12} - \log(A) - \frac{1}{240 {x}^{2}}

Monotonicity and convexity

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G(x)>0G(x) > 0
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(x>x0)    (G(n)(x)>0)   where x0={0,n=02.557664,n=11.898850,n=20.788740,n=3\left(x > {x}_{0}\right) \implies \left({G}^{(n)}(x) > 0\right)\; \text{ where } {x}_{0} = \begin{cases} 0, & n = 0\\2.557664, & n = 1\\1.898850, & n = 2\\0.788740, & n = 3\\ \end{cases}
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(x>x0)    (dndxn[logG(x)]>0)   where x0={3,n=02.557664,n=11.925864,n=20,n=3\left(x > {x}_{0}\right) \implies \left(\frac{d^{n}}{{d x}^{n}} \left[\log G(x)\right] > 0\right)\; \text{ where } {x}_{0} = \begin{cases} 3, & n = 0\\2.557664, & n = 1\\1.925864, & n = 2\\0, & n = 3\\ \end{cases}

Matrix formulas

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det ⁣((B0+0B0+1B0+nB1+0B1+1B1+nBn+0Bn+1Bn+n))=k=1nk!=G ⁣(n+2)\operatorname{det}\!\left(\displaystyle{\begin{pmatrix} B_{0 + 0} & B_{0 + 1} & \cdots & B_{0 + n} \\ B_{1 + 0} & B_{1 + 1} & \cdots & B_{1 + n} \\ \vdots & \vdots & \ddots & \vdots \\ B_{n + 0} & B_{n + 1} & \ldots & B_{n + n} \end{pmatrix}}\right) = \prod_{k=1}^{n} k ! = G\!\left(n + 2\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC