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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 — log⁡G(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 log⁡G(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 z∈CG(z) \text{ is holomorphic on } z \in \mathbb{C}
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n∈Z    ⟹    G(n)∈Z≥0n \in \mathbb{Z} \;\implies\; G(n) \in \mathbb{Z}_{\ge 0}
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x∈R    ⟹    G(x)∈Rx \in \mathbb{R} \;\implies\; G(x) \in \mathbb{R}
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z∈C    ⟹    G(z)∈Cz \in \mathbb{C} \;\implies\; G(z) \in \mathbb{C}

Logarithmic Barnes G-function

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log⁡G(z) is holomorphic on z∈C∖(−∞,0]\log G(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]
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x∈(0,∞)    ⟹    log⁡G(x)∈Rx \in \left(0, \infty\right) \;\implies\; \log G(x) \in \mathbb{R}
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z∈C∖{0,−1,…}    ⟹    log⁡G(z)∈Cz \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \log G(z) \in \mathbb{C}
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z∈{0,−1,…}    ⟹    log⁡G(z)∈{−∞}z \in \{0, -1, \ldots\} \;\implies\; \log G(z) \in \left\{-\infty\right\}

Logarithmic form

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G(z)=exp⁡ ⁣(log⁡G(z))G(z) = \exp\!\left(\log G(z)\right)
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log⁡G(x)={log⁡ ⁣(G(x)),x>0log⁡ ⁣(∣G(x)∣)+12n(n−1)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,∞)  or⁡  ∣z−2.5∣<2.5)    ⟹    (log⁡G(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=1n−2k!,n≥10,n≤0G(n) = \begin{cases} \prod_{k=1}^{n - 2} k !, & n \ge 1\\0, & n \le 0\\ \end{cases}
daef08
log⁡G(n)={log⁡ ⁣(G(n)),n≥1−∞,n≤0\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 0≤n≤150 \le n \le 15
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Table of G ⁣(10n)G\!\left({10}^{n}\right) to 50 digits for 0≤n≤100 \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}}
ce66a9
G ⁣(14)=e3/32−G/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)−12−γG'(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)+(n−1)(Hn−2−γ−1)+12),n≥2G'(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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zeros⁡ z∈CG(z)={0,−1,…}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} G(z) = \{0, -1, \ldots\}
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ord⁡z=−nG(z)=n+1\mathop{\operatorname{ord}}\limits_{z=-n} G(z) = n + 1
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zeros⁡ z∈Clog⁡G(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⁡ ⁣(log⁡G(z),z,C)={0,−1,…}\operatorname{BranchPoints}\!\left(\log G(z), z, \mathbb{C}\right) = \{0, -1, \ldots\}
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BranchCuts⁡ ⁣(log⁡G(z),z,C)={(−n−1,−n):n∈Z≥0}\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⁡ ⁣(log⁡G(x))=n(n−1)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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lim⁡ε→0+[log⁡G ⁣(x+εi)]=log⁡G(x)\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x + \varepsilon i\right)\right] = \log G(x)
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lim⁡ε→0+[log⁡G ⁣(x−εi)]=log⁡G(x)‾=log⁡G(x)−n(n−1)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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log⁡G ⁣(z+1)=log⁡Γ(z)+log⁡G(z)\log G\!\left(z + 1\right) = \log \Gamma(z) + \log G(z)
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G ⁣(z+n)=[∏k=1n(z+k−1)n−k](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 ⁣(1−x)=(−1)⌊(x−1)/2⌋+1G ⁣(1+x)(∣sin⁡ ⁣(πx)∣π)xexp⁡ ⁣(12πIm⁡ ⁣(Li⁡2 ⁣(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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log⁡G ⁣(1−x)=log⁡G ⁣(1+x)+xlog⁡ ⁣(∣sin⁡ ⁣(πx)∣π)+12πIm⁡ ⁣(Li⁡2 ⁣(e2πix))+sgn⁡(x)n(n+1)πi2   where n=⌊x⌋\log G\!\left(1 - x\right) = \log G\!\left(1 + x\right) + x \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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log⁡G ⁣(1−z)=log⁡G ⁣(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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log⁡G ⁣(1−z)=log⁡G ⁣(1+z)−log⁡ ⁣(2π)z+{∫0iπxcot⁡ ⁣(πx) dx+∫izπxcot⁡ ⁣(πx) dx,−1<Re⁡(z)<1  or⁡  Im⁡(z)>0  or⁡  (Im⁡(z)=0  and⁡  Re⁡(z)<1)∫0−iπxcot⁡ ⁣(πx) dx+∫−izπxcot⁡ ⁣(πx) dx,−1<Re⁡(z)<1  or⁡  Im⁡(z)<0  or⁡  (Im⁡(z)=0  and⁡  Re⁡(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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log⁡G ⁣(1−z)=log⁡G ⁣(1+z)+{F(z),0<Re⁡(z)<1  or⁡  Im⁡(z)>0  or⁡  (Im⁡(z)=0  and⁡  Re⁡(z)<1)−F ⁣(−z),−1<Re⁡(z)<0  or⁡  Im⁡(z)<0  or⁡  (Im⁡(z)=0  and⁡  Re⁡(z)>−1)   where F(z)=πi2(z2−z+16)−z(log⁡Γ(z)+log⁡Γ ⁣(1−z))−i2πLi⁡2 ⁣(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)(n2−1)nn2z2/2−nz+5/12(2π)(n−1)(1−nz)/2∏i=0n−1∏j=0n−1G ⁣(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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log⁡G ⁣(z‾)={log⁡G(z),z∈(−∞,0]log⁡G(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)((z−1)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 [log⁡G(z)]=(z−1)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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log⁡G(z)=(z−1)log⁡Γ(z)−ζ′ ⁣(−1,z)+ζ′(−1)\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)

Series and product representations

Taylor series

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log⁡G ⁣(1+z)=log⁡ ⁣(2π)−12z−1+γ2z2+∑n=3∞(−1)n+1ζ ⁣(n−1)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)/2∏k=1∞[(1+zk)kexp⁡ ⁣(z22k−z)]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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log⁡G ⁣(z+1)=z24+zlog⁡Γ ⁣(z+1)−(z(z+1)2+112)log⁡(z)−log⁡(A)+∑n=1N−1B2n+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)−∑k=0NB2k(2k)!t2k)e−ztt3 dtR_{N}\!\left(z\right) = \int_{0}^{\infty} \left(\frac{t}{2} \coth\!\left(\frac{t}{2}\right) - \sum_{k=0}^{N} \frac{B_{2 k}}{\left(2 k\right)!} {t}^{2 k}\right) \frac{{e}^{-z t}}{{t}^{3}} \, dt
092cee
RN ⁣(z)=1z2N(−1)N+1π∫0∞t2N−11+(tz)2Li⁡2 ⁣(e−2πt) 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)∫0∞B2N+1 ⁣(t−⌊t⌋)(t+z)2N dtR_{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+2∣2N(2N+1)(2N+2)∣z∣2N{1,∣arg⁡(z)∣≤π4sec⁡2N+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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log⁡G ⁣(z+1)=z(1−z)2+z2log⁡ ⁣(2π)+zlog⁡Γ(z)−∫0zlog⁡Γ(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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log⁡G ⁣(z+1)=z(1−z)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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log⁡G ⁣(z+1)=z24(2log⁡(z)−3)+zlog⁡ ⁣(2π)2+112−log⁡(A)−∫0∞xlog⁡ ⁣(x2+z2)e2πx−1 dx\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
6395ee
log⁡G ⁣(z+1)=zlog⁡Γ(z)+z24−log⁡(z)2B2 ⁣(z)−log⁡(A)−∫0∞e−zxx2(11−e−x−1x−12−x12) 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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log⁡G ⁣(x+1)<(x22−112)log⁡(x)−3x24+log⁡ ⁣(2π)2x+112−log⁡(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)
3544a0
log⁡G ⁣(x+1)>(x22−112)log⁡(x)−3x24+log⁡ ⁣(2π)2x+112−log⁡(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[log⁡G(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

dc6806
det⁡(B0+0B0+1⋯B0+nB1+0B1+1⋯B1+n⋮⋮⋱⋮Bn+0Bn+1…Bn+n)=∏k=1nk!=G ⁣(n+2)\operatorname{det}\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}} = \prod_{k=1}^{n} k ! = G\!\left(n + 2\right)
f4750b
det⁡ ⁣(Hn)=(G ⁣(n+1))4G ⁣(2n+1)\operatorname{det}\!\left(H_{n}\right) = \frac{{\left(G\!\left(n + 1\right)\right)}^{4}}{G\!\left(2 n + 1\right)}

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

2021-03-15 19:12:00.328586 UTC