# Barnes G-function

## Definitions

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

## Illustrations

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

## Domain

### Barnes G-function

$G(z) \text{ is holomorphic on } z \in \mathbb{C}$
$n \in \mathbb{Z} \;\implies\; G(n) \in \mathbb{Z}_{\ge 0}$
$x \in \mathbb{R} \;\implies\; G(x) \in \mathbb{R}$
$z \in \mathbb{C} \;\implies\; G(z) \in \mathbb{C}$

### Logarithmic Barnes G-function

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

## Logarithmic form

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

## Specific values

### Integers

$G(n) = \begin{cases} \prod_{k=1}^{n - 2} k !, & n \ge 1\\0, & n \le 0\\ \end{cases}$
$\log G(n) = \begin{cases} \log\!\left(G(n)\right), & n \ge 1\\-\infty, & n \le 0\\ \end{cases}$
Table of $G(n)$ for $0 \le n \le 15$
Table of $G\!\left({10}^{n}\right)$ to 50 digits for $0 \le n \le 10$

### Rational arguments

$G\!\left(\frac{1}{2}\right) = \frac{{2}^{1 / 24} {e}^{1 / 8}}{{\pi}^{1 / 4} {A}^{3 / 2}}$
$G\!\left(\frac{1}{4}\right) = \frac{{e}^{3 / 32 - G / 4 \pi}}{{A}^{9 / 8} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{3 / 4}}$
$G\!\left(\frac{3}{4}\right) = \frac{{e}^{3 / 32 + G / 4 \pi} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{1 / 4}}{{2}^{1 / 8} {\pi}^{1 / 4} {A}^{9 / 8}}$

### Derivatives

$G'(0) = 1$
$G'(1) = \frac{\log\!\left(2 \pi\right) - 1}{2}$
$G'(2) = \frac{\log\!\left(2 \pi\right) - 1}{2} - \gamma$
$G'(n) = \begin{cases} 0, & n < 0\\1, & n = 0\\\frac{1}{2} \left(\log\!\left(2 \pi\right) - 1\right), & n = 1\\G(n) \left(\frac{1}{2} \log\!\left(2 \pi\right) + \left(n - 1\right) \left(H_{n - 2} - \gamma - 1\right) + \frac{1}{2}\right), & n \ge 2\\ \end{cases}$

## Singularities and zeros

### Zeros

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

### Branch cuts

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

## Functional equations

### Recurrence relation

$G\!\left(z + 1\right) = \Gamma(z) G(z)$
$\log G\!\left(z + 1\right) = \log \Gamma(z) + \log G(z)$
$G\!\left(z + n\right) = \left[\prod_{k=1}^{n} {\left(z + k - 1\right)}^{n - k}\right] {\left(\Gamma(z)\right)}^{n} G(z)$

### Reflection formula, real variables

$G\!\left(1 - x\right) = {\left(-1\right)}^{\left\lfloor \left( x - 1 \right) / 2 \right\rfloor + 1} G\!\left(1 + x\right) {\left(\frac{\left|\sin\!\left(\pi x\right)\right|}{\pi}\right)}^{x} \exp\!\left(\frac{1}{2 \pi} \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left({e}^{2 \pi i x}\right)\right)\right)$
$\log G\!\left(1 - x\right) = \log G\!\left(1 + x\right) + x \log\!\left(\frac{\left|\sin\!\left(\pi x\right)\right|}{\pi}\right) + \frac{1}{2 \pi} \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left({e}^{2 \pi i x}\right)\right) + \operatorname{sgn}(x) n \left(n + 1\right) \frac{\pi i}{2}\; \text{ where } n = \left\lfloor x \right\rfloor$

### Reflection formula, complex variables

$\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) - \log\!\left(2 \pi\right) z + \int_{0}^{z} \pi x \cot\!\left(\pi x\right) \, dx$
$\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) - \log\!\left(2 \pi\right) z + \begin{cases} \int_{0}^{i} \pi x \cot\!\left(\pi x\right) \, dx + \int_{i}^{z} \pi x \cot\!\left(\pi x\right) \, dx, & -1 < \operatorname{Re}(z) < 1 \;\mathbin{\operatorname{or}}\; \operatorname{Im}(z) > 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) < 1\right)\\\int_{0}^{-i} \pi x \cot\!\left(\pi x\right) \, dx + \int_{-i}^{z} \pi x \cot\!\left(\pi x\right) \, dx, & -1 < \operatorname{Re}(z) < 1 \;\mathbin{\operatorname{or}}\; \operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > -1\right)\\ \end{cases}$
$\log G\!\left(1 - z\right) = \log G\!\left(1 + z\right) + \begin{cases} F(z), & 0 < \operatorname{Re}(z) < 1 \;\mathbin{\operatorname{or}}\; \operatorname{Im}(z) > 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) < 1\right)\\-F\!\left(-z\right), & -1 < \operatorname{Re}(z) < 0 \;\mathbin{\operatorname{or}}\; \operatorname{Im}(z) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > -1\right)\\ \end{cases}\; \text{ where } F(z) = \frac{\pi i}{2} \left({z}^{2} - z + \frac{1}{6}\right) - z \left(\log \Gamma(z) + \log \Gamma\!\left(1 - z\right)\right) - \frac{i}{2 \pi} \operatorname{Li}_{2}\!\left({e}^{2 \pi i z}\right)$

### Multiplication theorem

$G\!\left(n z\right) = {e}^{\left(\log(A) - 1 / 12\right) \left({n}^{2} - 1\right)} {n}^{{n}^{2} {z}^{2} / 2 - n z + 5 / 12} {\left(2 \pi\right)}^{\left(n - 1\right) \left(1 - n z\right) / 2} \prod_{i=0}^{n - 1} \prod_{j=0}^{n - 1} G\!\left(z + \frac{i + j}{n}\right)$

### Conjugate symmetry

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

## Derivatives and differential equations

$G'(z) = G(z) \left(\left(z - 1\right) \psi\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right) + 1}{2}\right)$
$\frac{d}{d z}\, \left[\log G(z)\right] = \left(z - 1\right) \psi\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right) + 1}{2}$

## Representation by other functions

$\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)$

## Series and product representations

### Taylor series

$\log G\!\left(1 + z\right) = \frac{\log\!\left(2 \pi\right) - 1}{2} z - \frac{1 + \gamma}{2} {z}^{2} + \sum_{n=3}^{\infty} \frac{{\left(-1\right)}^{n + 1} \zeta\!\left(n - 1\right)}{n} {z}^{n}$

### Weierstrass product

$G\!\left(z + 1\right) = {\left(2 \pi\right)}^{z / 2} {e}^{-\left( z + \left(\gamma + 1\right) {z}^{2} \right) / 2} \prod_{k=1}^{\infty} \left[{\left(1 + \frac{z}{k}\right)}^{k} \exp\!\left(\frac{{z}^{2}}{2 k} - z\right)\right]$

### Asymptotic expansion

Symbol: LogBarnesGRemainder $R_{N}\!\left(z\right)$ Remainder term in asymptotic expansion of logarithmic Barnes G-function
$\log G\!\left(z + 1\right) = \frac{{z}^{2}}{4} + z \log \Gamma\!\left(z + 1\right) - \left(\frac{z \left(z + 1\right)}{2} + \frac{1}{12}\right) \log(z) - \log(A) + \sum_{n=1}^{N - 1} \frac{B_{2 n + 2}}{2 n \left(2 n + 1\right) \left(2 n + 2\right) {z}^{2 n}} + R_{N}\!\left(z\right)$
$R_{N}\!\left(z\right) = \int_{0}^{\infty} \left(\frac{t}{2} \coth\!\left(\frac{t}{2}\right) - \sum_{k=0}^{N} \frac{B_{2 k}}{\left(2 k\right)!} {t}^{2 k}\right) \frac{{e}^{-z t}}{{t}^{3}} \, dt$
$R_{N}\!\left(z\right) = \frac{1}{{z}^{2 N}} \frac{{\left(-1\right)}^{N + 1}}{\pi} \int_{0}^{\infty} \frac{{t}^{2 N - 1}}{1 + {\left(\frac{t}{z}\right)}^{2}} \operatorname{Li}_{2}\!\left({e}^{-2 \pi t}\right) \, dt$
$R_{N}\!\left(z\right) = \frac{1}{2 N \left(2 N + 1\right)} \int_{0}^{\infty} \frac{B_{2 N + 1}\!\left(t - \left\lfloor t \right\rfloor\right)}{{\left(t + z\right)}^{2 N}} \, dt$
$\left|R_{N}\!\left(z\right)\right| \le \frac{\left|B_{2 N + 2}\right|}{2 N \left(2 N + 1\right) \left(2 N + 2\right) {\left|z\right|}^{2 N}} \begin{cases} 1, & \left|\arg(z)\right| \le \frac{\pi}{4}\\\sec^{2 N + 1}\!\left(\frac{1}{2} \arg(z)\right), & \left|\arg(z)\right| < \pi\\ \end{cases}$

## Integral representations

$\log G\!\left(z + 1\right) = \frac{z \left(1 - z\right)}{2} + \frac{z}{2} \log\!\left(2 \pi\right) + z \log \Gamma(z) - \int_{0}^{z} \log \Gamma(x) \, dx$
$\log G\!\left(z + 1\right) = \frac{z \left(1 - z\right)}{2} + \frac{z}{2} \log\!\left(2 \pi\right) + \int_{0}^{z} x \psi\!\left(x\right) \, dx$
$\log G\!\left(z + 1\right) = \frac{{z}^{2}}{4} \left(2 \log(z) - 3\right) + \frac{z \log\!\left(2 \pi\right)}{2} + \frac{1}{12} - \log(A) - \int_{0}^{\infty} \frac{x \log\!\left({x}^{2} + {z}^{2}\right)}{{e}^{2 \pi x} - 1} \, dx$
$\log G\!\left(z + 1\right) = z \log \Gamma(z) + \frac{{z}^{2}}{4} - \frac{\log(z)}{2} B_{2}\!\left(z\right) - \log(A) - \int_{0}^{\infty} \frac{{e}^{-z x}}{{x}^{2}} \left(\frac{1}{1 - {e}^{-x}} - \frac{1}{x} - \frac{1}{2} - \frac{x}{12}\right) \, dx$

## Bounds and inequalities

### Upper and lower bounds

$\log G\!\left(x + 1\right) < \left(\frac{{x}^{2}}{2} - \frac{1}{12}\right) \log(x) - \frac{3 {x}^{2}}{4} + \frac{\log\!\left(2 \pi\right)}{2} x + \frac{1}{12} - \log(A)$
$\log G\!\left(x + 1\right) > \left(\frac{{x}^{2}}{2} - \frac{1}{12}\right) \log(x) - \frac{3 {x}^{2}}{4} + \frac{\log\!\left(2 \pi\right)}{2} x + \frac{1}{12} - \log(A) - \frac{1}{240 {x}^{2}}$

### Monotonicity and convexity

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

## Matrix formulas

$\operatorname{det}\displaystyle{\begin{pmatrix} B_{0 + 0} & B_{0 + 1} & \cdots & B_{0 + n} \\ B_{1 + 0} & B_{1 + 1} & \cdots & B_{1 + n} \\ \vdots & \vdots & \ddots & \vdots \\ B_{n + 0} & B_{n + 1} & \ldots & B_{n + n} \end{pmatrix}} = \prod_{k=1}^{n} k ! = G\!\left(n + 2\right)$
$\operatorname{det}\!\left(H_{n}\right) = \frac{{\left(G\!\left(n + 1\right)\right)}^{4}}{G\!\left(2 n + 1\right)}$

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