Barnes G-function

Symbol: BarnesG —

G(z)

— Barnes G-function

BarnesG(z), rendered as

G(z)

, represents the Barnes G-function of argument

z

.

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function

Source code for this entry:

Entry(ID("82a1b1"),
    SymbolDefinition(BarnesG, BarnesG(z), "Barnes G-function"),
    Description(SourceForm(BarnesG(z)), ", rendered as", BarnesG(z), ", represents the Barnes G-function of argument", z, "."))

da1509

Symbol: LogBarnesG —

\log G(z)

— Logarithmic Barnes G-function

LogBarnesG(z), rendered as

\log G(z)

, represents the logarithmic Barnes G-function of argument

z

.

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function

Source code for this entry:

Entry(ID("da1509"),
    SymbolDefinition(LogBarnesG, LogBarnesG(z), "Logarithmic Barnes G-function"),
    Description(SourceForm(LogBarnesG(z)), ", rendered as", LogBarnesG(z), ", represents the logarithmic Barnes G-function of argument", z, "."))

488c5c

Image: Plot of

G(x)

on

x \in \left[-4, 6\right]

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("488c5c"),
    Image(Description("Plot of", BarnesG(x), "on", Element(x, ClosedInterval(-4, 6))), ImageSource("plot_barnes_g")))

106bf7

Image: X-ray of

G(z)

on

z \in \left[-4, 6\right] + \left[-5, 5\right] i

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("106bf7"),
    Image(Description("X-ray of", BarnesG(z), "on", Element(z, Add(ClosedInterval(-4, 6), Mul(ClosedInterval(-5, 5), ConstI)))), ImageSource("xray_barnes_g")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

e1497f

Image: X-ray of

\log G(z)

on

z \in \left[-4, 6\right] + \left[-5, 5\right] i

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("e1497f"),
    Image(Description("X-ray of", LogBarnesG(z), "on", Element(z, Add(ClosedInterval(-4, 6), Mul(ClosedInterval(-5, 5), ConstI)))), ImageSource("xray_log_barnes_g")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

971881

G(z) \text{ is holomorphic on } z \in \mathbb{C}

TeX:

G(z) \text{ is holomorphic on } z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`BarnesG`	$G(z)$	Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("971881"),
    Formula(IsHolomorphic(BarnesG(z), ForElement(z, CC))))

3d46ea

n \in \mathbb{Z} \;\implies\; G(n) \in \mathbb{Z}_{\ge 0}

TeX:

n \in \mathbb{Z} \;\implies\; G(n) \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers
`BarnesG`	$G(z)$	Barnes G-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("3d46ea"),
    Formula(Implies(Element(n, ZZ), Element(BarnesG(n), ZZGreaterEqual(0)))),
    Variables(n))

f3f0a7

x \in \mathbb{R} \;\implies\; G(x) \in \mathbb{R}

TeX:

x \in \mathbb{R} \;\implies\; G(x) \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`RR`	$\mathbb{R}$	Real numbers
`BarnesG`	$G(z)$	Barnes G-function

Source code for this entry:

Entry(ID("f3f0a7"),
    Formula(Implies(Element(x, RR), Element(BarnesG(x), RR))),
    Variables(x))

8ec739

z \in \mathbb{C} \;\implies\; G(z) \in \mathbb{C}

TeX:

z \in \mathbb{C} \;\implies\; G(z) \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`BarnesG`	$G(z)$	Barnes G-function

Source code for this entry:

Entry(ID("8ec739"),
    Formula(Implies(Element(z, CC), Element(BarnesG(z), CC))),
    Variables(z))

037342

\log G(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]

TeX:

\log G(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("037342"),
    Formula(IsHolomorphic(LogBarnesG(z), ForElement(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))),
    Variables(z))

19b40f

x \in \left(0, \infty\right) \;\implies\; \log G(x) \in \mathbb{R}

TeX:

x \in \left(0, \infty\right) \;\implies\; \log G(x) \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("19b40f"),
    Formula(Implies(Element(x, OpenInterval(0, Infinity)), Element(LogBarnesG(x), RR))),
    Variables(x))

a471d0

z \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \log G(z) \in \mathbb{C}

TeX:

z \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \log G(z) \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function

Source code for this entry:

Entry(ID("a471d0"),
    Formula(Implies(Element(z, SetMinus(CC, ZZLessEqual(0))), Element(LogBarnesG(z), CC))),
    Variables(z))

62bdb8

z \in \{0, -1, \ldots\} \;\implies\; \log G(z) \in \left\{-\infty\right\}

TeX:

z \in \{0, -1, \ldots\} \;\implies\; \log G(z) \in \left\{-\infty\right\}

Definitions:

Fungrim symbol	Notation	Short description
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("62bdb8"),
    Formula(Implies(Element(z, ZZLessEqual(0)), Element(LogBarnesG(z), Set(Neg(Infinity))))),
    Variables(z))

b4355e

G(z) = \exp\!\left(\log G(z)\right)

Assumptions:

z \in \mathbb{C}

TeX:

G(z) = \exp\!\left(\log G(z)\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Exp`	${e}^{z}$	Exponential function
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b4355e"),
    Formula(Equal(BarnesG(z), Exp(LogBarnesG(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

5a11eb

\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

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \{0, -1, \ldots\}

TeX:

\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

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`BarnesG`	$G(z)$	Barnes G-function
`Abs`	$\left\|z\right\|$	Absolute value
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("5a11eb"),
    Formula(Equal(LogBarnesG(x), Where(Cases(Tuple(Log(BarnesG(x)), Greater(x, 0)), Tuple(Add(Log(Abs(BarnesG(x))), Mul(Mul(Mul(Mul(Div(1, 2), n), Sub(n, 1)), Pi), ConstI)), Otherwise)), Equal(n, Floor(x))))),
    Variables(x),
    Assumptions(And(Element(x, RR), NotElement(x, ZZLessEqual(0)))))

d9a7a3

\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)

Assumptions:

z \in \mathbb{C}

TeX:

\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)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`Abs`	$\left\|z\right\|$	Absolute value
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`BarnesG`	$G(z)$	Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d9a7a3"),
    Formula(Implies(Or(Element(z, OpenInterval(0, Infinity)), Less(Abs(Sub(z, Decimal("2.5"))), Decimal("2.5"))), Equal(LogBarnesG(z), Log(BarnesG(z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

33f13a

G(n) = \begin{cases} \prod_{k=1}^{n - 2} k !, & n \ge 1\\0, & n \le 0\\ \end{cases}

Assumptions:

n \in \mathbb{Z}

TeX:

G(n) = \begin{cases} \prod_{k=1}^{n - 2} k !, & n \ge 1\\0, & n \le 0\\ \end{cases}

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Product`	$\prod_{n} f(n)$	Product
`Factorial`	$n !$	Factorial
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("33f13a"),
    Formula(Equal(BarnesG(n), Cases(Tuple(Product(Factorial(k), For(k, 1, Sub(n, 2))), GreaterEqual(n, 1)), Tuple(0, LessEqual(n, 0))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

daef08

\log G(n) = \begin{cases} \log\!\left(G(n)\right), & n \ge 1\\-\infty, & n \le 0\\ \end{cases}

Assumptions:

n \in \mathbb{Z}

TeX:

\log G(n) = \begin{cases} \log\!\left(G(n)\right), & n \ge 1\\-\infty, & n \le 0\\ \end{cases}

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`BarnesG`	$G(z)$	Barnes G-function
`Infinity`	$\infty$	Positive infinity
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("daef08"),
    Formula(Equal(LogBarnesG(n), Cases(Tuple(Log(BarnesG(n)), GreaterEqual(n, 1)), Tuple(Neg(Infinity), LessEqual(n, 0))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

5cb675

Table of

G(n)

for

0 \le n \le 15

$n$	$G(n)$
0	0
1	1
2	1
3	1
4	2
5	12
6	288
7	34560
8	24883200
9	125411328000
10	5056584744960000
11	1834933472251084800000
12	6658606584104736522240000000
13	265790267296391946810949632000000000
14	127313963299399416749559771247411200000000000
15	792786697595796795607377086400871488552960000000000000

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function

Source code for this entry:

Entry(ID("5cb675"),
    Description("Table of", BarnesG(n), "for", LessEqual(0, n, 15)),
    Table(Var(n), TableValueHeadings(n, BarnesG(n)), TableSplit(1), List(Tuple(0, 0), Tuple(1, 1), Tuple(2, 1), Tuple(3, 1), Tuple(4, 2), Tuple(5, 12), Tuple(6, 288), Tuple(7, 34560), Tuple(8, 24883200), Tuple(9, 125411328000), Tuple(10, 5056584744960000), Tuple(11, 1834933472251084800000), Tuple(12, 6658606584104736522240000000), Tuple(13, 265790267296391946810949632000000000), Tuple(14, 127313963299399416749559771247411200000000000), Tuple(15, 792786697595796795607377086400871488552960000000000000))))

dbc117

Table of

G\!\left({10}^{n}\right)

to 50 digits for

0 \le n \le 10

$n$	$G\!\left({10}^{n}\right) \; (\text{nearest } 50 \text{D})$
0	1
1	5056584744960000
2	3.1036100626369830847879402668101769402280611882492 · 10⁶⁶²⁶
3	2.0045690761252153894020068969764708165025223902021 · 10^1172113
4	7.8913000980387915476627721291099032636382430763247 · 10^167396248
5	6.0203407218068785584794013164002058716523147916958 · 10^21742374725
6	1.0145655480290378725684376810089100792606395451713 · 10^{2674273971959}
7	5.2418488985408575463326646933057662934049190054866 · 10^{317427852191102}
8	5.7976150706924830557487583722980527113426519510559 · 10^{36742790669064055}
9	4.1978917865925966071745800658793636650835499429177 · 10^{4174279130405945548}
10	4.8323663133177075948431502316007152775466606093624 · 10^{467427913765589957090}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("dbc117"),
    Description("Table of", BarnesG(Pow(10, n)), "to 50 digits for", LessEqual(0, n, 10)),
    Table(Var(n), TableValueHeadings(n, NearestDecimal(BarnesG(Pow(10, n)), 50)), TableSplit(1), List(Tuple(0, Decimal("1")), Tuple(1, Decimal("5056584744960000")), Tuple(2, Decimal("3.1036100626369830847879402668101769402280611882492e+6626")), Tuple(3, Decimal("2.0045690761252153894020068969764708165025223902021e+1172113")), Tuple(4, Decimal("7.8913000980387915476627721291099032636382430763247e+167396248")), Tuple(5, Decimal("6.0203407218068785584794013164002058716523147916958e+21742374725")), Tuple(6, Decimal("1.0145655480290378725684376810089100792606395451713e+2674273971959")), Tuple(7, Decimal("5.2418488985408575463326646933057662934049190054866e+317427852191102")), Tuple(8, Decimal("5.7976150706924830557487583722980527113426519510559e+36742790669064055")), Tuple(9, Decimal("4.1978917865925966071745800658793636650835499429177e+4174279130405945548")), Tuple(10, Decimal("4.8323663133177075948431502316007152775466606093624e+467427913765589957090")))))

8b7991

G\!\left(\frac{1}{2}\right) = \frac{{2}^{1 / 24} {e}^{1 / 8}}{{\pi}^{1 / 4} {A}^{3 / 2}}

TeX:

G\!\left(\frac{1}{2}\right) = \frac{{2}^{1 / 24} {e}^{1 / 8}}{{\pi}^{1 / 4} {A}^{3 / 2}}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("8b7991"),
    Formula(Equal(BarnesG(Div(1, 2)), Div(Mul(Pow(2, Div(1, 24)), Exp(Div(1, 8))), Mul(Pow(Pi, Div(1, 4)), Pow(ConstGlaisher, Div(3, 2)))))))

ce66a9

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}}

TeX:

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}}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Pow`	${a}^{b}$	Power
`ConstE`	$e$	The constant e (2.718...)
`ConstCatalan`	$G$	Catalan's constant
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("ce66a9"),
    Formula(Equal(BarnesG(Div(1, 4)), Div(Pow(ConstE, Sub(Div(3, 32), Div(ConstCatalan, Mul(4, Pi)))), Mul(Pow(ConstGlaisher, Div(9, 8)), Pow(Gamma(Div(1, 4)), Div(3, 4)))))))

dc507f

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}}

TeX:

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}}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Pow`	${a}^{b}$	Power
`ConstE`	$e$	The constant e (2.718...)
`ConstCatalan`	$G$	Catalan's constant
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("dc507f"),
    Formula(Equal(BarnesG(Div(3, 4)), Div(Mul(Pow(ConstE, Add(Div(3, 32), Div(ConstCatalan, Mul(4, Pi)))), Pow(Gamma(Div(1, 4)), Div(1, 4))), Mul(Mul(Pow(2, Div(1, 8)), Pow(Pi, Div(1, 4))), Pow(ConstGlaisher, Div(9, 8)))))))

90b367

G'(0) = 1

TeX:

G'(0) = 1

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`BarnesG`	$G(z)$	Barnes G-function

Source code for this entry:

Entry(ID("90b367"),
    Formula(Equal(ComplexDerivative(BarnesG(z), For(z, 0)), 1)))

dbfd5b

G'(1) = \frac{\log\!\left(2 \pi\right) - 1}{2}

TeX:

G'(1) = \frac{\log\!\left(2 \pi\right) - 1}{2}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`BarnesG`	$G(z)$	Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("dbfd5b"),
    Formula(Equal(ComplexDerivative(BarnesG(z), For(z, 1)), Div(Sub(Log(Mul(2, Pi)), 1), 2))))

a54fb0

G'(2) = \frac{\log\!\left(2 \pi\right) - 1}{2} - \gamma

TeX:

G'(2) = \frac{\log\!\left(2 \pi\right) - 1}{2} - \gamma

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`BarnesG`	$G(z)$	Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstGamma`	$\gamma$	The constant gamma (0.577...)

Source code for this entry:

Entry(ID("a54fb0"),
    Formula(Equal(ComplexDerivative(BarnesG(z), For(z, 2)), Sub(Div(Sub(Log(Mul(2, Pi)), 1), 2), ConstGamma))))

f50c74

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}

Assumptions:

n \in \mathbb{Z}

TeX:

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}

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`BarnesG`	$G(z)$	Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("f50c74"),
    Formula(Equal(ComplexDerivative(BarnesG(z), For(z, n)), Cases(Tuple(0, Less(n, 0)), Tuple(1, Equal(n, 0)), Tuple(Mul(Div(1, 2), Sub(Log(Mul(2, Pi)), 1)), Equal(n, 1)), Tuple(Mul(BarnesG(n), Add(Add(Mul(Div(1, 2), Log(Mul(2, Pi))), Mul(Sub(n, 1), Sub(Sub(HarmonicNumber(Sub(n, 2)), ConstGamma), 1))), Div(1, 2))), GreaterEqual(n, 2))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

e488dc

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} G(z) = \{0, -1, \ldots\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} G(z) = \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`BarnesG`	$G(z)$	Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("e488dc"),
    Formula(Equal(Zeros(BarnesG(z), ForElement(z, CC)), ZZLessEqual(0))))

f77124

\mathop{\operatorname{ord}}\limits_{z=-n} G(z) = n + 1

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\mathop{\operatorname{ord}}\limits_{z=-n} G(z) = n + 1

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexZeroMultiplicity`	$\mathop{\operatorname{ord}}\limits_{z=c} f(z)$	Multiplicity (order) of complex zero
`BarnesG`	$G(z)$	Barnes G-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("f77124"),
    Formula(Equal(ComplexZeroMultiplicity(BarnesG(z), For(z, Neg(n))), Add(n, 1))),
    Variables(n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)))))

2b021c

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log G(z) = \left\{1, 2, 3\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log G(z) = \left\{1, 2, 3\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("2b021c"),
    Formula(Equal(Zeros(LogBarnesG(z), ForElement(z, CC)), Set(1, 2, 3))))

c3e340

\operatorname{BranchPoints}\!\left(\log G(z), z, \mathbb{C}\right) = \{0, -1, \ldots\}

TeX:

\operatorname{BranchPoints}\!\left(\log G(z), z, \mathbb{C}\right) = \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("c3e340"),
    Formula(Equal(BranchPoints(LogBarnesG(z), z, CC), ZZLessEqual(0))))

e1f73d

\operatorname{BranchCuts}\!\left(\log G(z), z, \mathbb{C}\right) = \left\{ \left(-n - 1, -n\right) : n \in \mathbb{Z}_{\ge 0} \right\}

TeX:

\operatorname{BranchCuts}\!\left(\log G(z), z, \mathbb{C}\right) = \left\{ \left(-n - 1, -n\right) : n \in \mathbb{Z}_{\ge 0} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e1f73d"),
    Formula(Equal(BranchCuts(LogBarnesG(z), z, CC), Set(OpenInterval(Sub(Neg(n), 1), Neg(n)), ForElement(n, ZZGreaterEqual(0))))))

a044e1

\operatorname{Im}\!\left(\log G(x)\right) = \frac{n \left(n - 1\right)}{2} \pi\; \text{ where } n = \left\lfloor x \right\rfloor

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

TeX:

\operatorname{Im}\!\left(\log G(x)\right) = \frac{n \left(n - 1\right)}{2} \pi\; \text{ where } n = \left\lfloor x \right\rfloor

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}(z)$	Imaginary part
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Pi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("a044e1"),
    Formula(Equal(Im(LogBarnesG(x)), Where(Mul(Div(Mul(n, Sub(n, 1)), 2), Pi), Equal(n, Floor(x))))),
    Variables(x),
    Assumptions(And(Element(x, RR), Less(x, 0), NotElement(x, ZZ))))

cc3a51

\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x + \varepsilon i\right)\right] = \log G(x)

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

TeX:

\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x + \varepsilon i\right)\right] = \log G(x)

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`RightLimit`	$\lim_{x \to {a}^{+}} f(x)$	Limiting value, from the right
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("cc3a51"),
    Formula(Equal(RightLimit(Brackets(LogBarnesG(Add(x, Mul(epsilon, ConstI)))), For(epsilon, 0)), LogBarnesG(x))),
    Variables(x),
    Assumptions(And(Element(x, RR), Less(x, 0), NotElement(x, ZZ))))

d35c54

\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

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

TeX:

\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

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`RightLimit`	$\lim_{x \to {a}^{+}} f(x)$	Limiting value, from the right
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`ConstI`	$i$	Imaginary unit
`Conjugate`	$\overline{z}$	Complex conjugate
`Pi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d35c54"),
    Formula(Equal(RightLimit(Brackets(LogBarnesG(Sub(x, Mul(epsilon, ConstI)))), For(epsilon, 0)), Conjugate(LogBarnesG(x)), Where(Sub(LogBarnesG(x), Mul(Mul(Mul(n, Sub(n, 1)), Pi), ConstI)), Equal(n, Floor(x))))),
    Variables(x),
    Assumptions(And(Element(x, RR), Less(x, 0), NotElement(x, ZZ))))

86b3ec

G\!\left(z + 1\right) = \Gamma(z) G(z)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

G\!\left(z + 1\right) = \Gamma(z) G(z)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("86b3ec"),
    Formula(Equal(BarnesG(Add(z, 1)), Mul(Gamma(z), BarnesG(z)))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

5261e3

\log G\!\left(z + 1\right) = \log \Gamma(z) + \log G(z)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

\log G\!\left(z + 1\right) = \log \Gamma(z) + \log G(z)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("5261e3"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Add(LogGamma(z), LogBarnesG(z)))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

7a36e5

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)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

TeX:

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)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7a36e5"),
    Formula(Equal(BarnesG(Add(z, n)), Mul(Mul(Brackets(Product(Pow(Sub(Add(z, k), 1), Sub(n, k)), For(k, 1, n))), Pow(Gamma(z), n)), BarnesG(z)))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)), Element(n, ZZGreaterEqual(0)))))

541e2e

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)

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \{-1, -2, \ldots\}

References:

https://doi.org/10.1145/384101.384104

TeX:

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)

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \{-1, -2, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Pow`	${a}^{b}$	Power
`Abs`	$\left\|z\right\|$	Absolute value
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`Im`	$\operatorname{Im}(z)$	Imaginary part
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("541e2e"),
    Formula(Equal(BarnesG(Sub(1, x)), Mul(Mul(Mul(Pow(-1, Add(Floor(Div(Sub(x, 1), 2)), 1)), BarnesG(Add(1, x))), Pow(Div(Abs(Sin(Mul(Pi, x))), Pi), x)), Exp(Mul(Div(1, Mul(2, Pi)), Im(PolyLog(2, Exp(Mul(Mul(Mul(2, Pi), ConstI), x))))))))),
    Variables(x),
    Assumptions(And(Element(x, RR), NotElement(x, ZZLessEqual(-1)))),
    References("https://doi.org/10.1145/384101.384104"))

d1a0ec

\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

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

TeX:

\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

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Abs`	$\left\|z\right\|$	Absolute value
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Sign`	$\operatorname{sgn}(z)$	Sign function
`RR`	$\mathbb{R}$	Real numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d1a0ec"),
    Formula(Equal(LogBarnesG(Sub(1, x)), Add(Add(Add(LogBarnesG(Add(1, x)), Mul(x, Log(Div(Abs(Sin(Mul(Pi, x))), Pi)))), Mul(Div(1, Mul(2, Pi)), Im(PolyLog(2, Exp(Mul(Mul(Mul(2, Pi), ConstI), x)))))), Where(Mul(Mul(Sign(x), Mul(n, Add(n, 1))), Div(Mul(Pi, ConstI), 2)), Equal(n, Floor(x)))))),
    Variables(x),
    Assumptions(And(Element(x, RR), NotElement(x, ZZ))))

b6017f

\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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)

TeX:

\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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("b6017f"),
    Formula(Equal(LogBarnesG(Sub(1, z)), Add(Sub(LogBarnesG(Add(1, z)), Mul(Log(Mul(2, Pi)), z)), Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, 0, z))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

23ed69

\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}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \mathbb{Z}

TeX:

\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}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`ConstI`	$i$	Imaginary unit
`Re`	$\operatorname{Re}(z)$	Real part
`Im`	$\operatorname{Im}(z)$	Imaginary part
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("23ed69"),
    Formula(Equal(LogBarnesG(Sub(1, z)), Add(Sub(LogBarnesG(Add(1, z)), Mul(Log(Mul(2, Pi)), z)), Cases(Tuple(Add(Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, 0, ConstI)), Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, ConstI, z))), Or(Less(-1, Re(z), 1), Greater(Im(z), 0), And(Equal(Im(z), 0), Less(Re(z), 1)))), Tuple(Add(Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, 0, Neg(ConstI))), Integral(Mul(Mul(Pi, x), Cot(Mul(Pi, x))), For(x, Neg(ConstI), z))), Or(Less(-1, Re(z), 1), Less(Im(z), 0), And(Equal(Im(z), 0), Greater(Re(z), -1)))))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZ))))

82b410

\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)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \mathbb{Z}

TeX:

\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)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Re`	$\operatorname{Re}(z)$	Real part
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("82b410"),
    Formula(Equal(LogBarnesG(Sub(1, z)), Add(LogBarnesG(Add(1, z)), Where(Cases(Tuple(F(z), Or(Less(0, Re(z), 1), Greater(Im(z), 0), And(Equal(Im(z), 0), Less(Re(z), 1)))), Tuple(Neg(F(Neg(z))), Or(Less(-1, Re(z), 0), Less(Im(z), 0), And(Equal(Im(z), 0), Greater(Re(z), -1))))), Equal(F(z), Sub(Sub(Mul(Div(Mul(Pi, ConstI), 2), Add(Sub(Pow(z, 2), z), Div(1, 6))), Mul(z, Add(LogGamma(z), LogGamma(Sub(1, z))))), Mul(Div(ConstI, Mul(2, Pi)), PolyLog(2, Exp(Mul(Mul(Mul(2, Pi), ConstI), z)))))))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZ))))

ea26d4

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)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

References:

https://arxiv.org/abs/math/0308086

TeX:

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)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Pow`	${a}^{b}$	Power
`ConstE`	$e$	The constant e (2.718...)
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Product`	$\prod_{n} f(n)$	Product
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ea26d4"),
    Formula(Equal(BarnesG(Mul(n, z)), Mul(Mul(Mul(Pow(ConstE, Mul(Sub(Log(ConstGlaisher), Div(1, 12)), Sub(Pow(n, 2), 1))), Pow(n, Add(Sub(Div(Mul(Pow(n, 2), Pow(z, 2)), 2), Mul(n, z)), Div(5, 12)))), Pow(Mul(2, Pi), Div(Mul(Sub(n, 1), Sub(1, Mul(n, z))), 2))), Product(Product(BarnesG(Add(z, Div(Add(i, j), n))), For(j, 0, Sub(n, 1))), For(i, 0, Sub(n, 1)))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(1)))),
    References("https://arxiv.org/abs/math/0308086"))

147db6

G\!\left(\overline{z}\right) = \overline{G(z)}

Assumptions:

z \in \mathbb{C}

TeX:

G\!\left(\overline{z}\right) = \overline{G(z)}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Conjugate`	$\overline{z}$	Complex conjugate
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("147db6"),
    Formula(Equal(BarnesG(Conjugate(z)), Conjugate(BarnesG(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

6c6d3e

\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}

Assumptions:

z \in \mathbb{C}

TeX:

\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}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Conjugate`	$\overline{z}$	Complex conjugate
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("6c6d3e"),
    Formula(Equal(LogBarnesG(Conjugate(z)), Cases(Tuple(LogBarnesG(z), Element(z, OpenClosedInterval(Neg(Infinity), 0))), Tuple(Conjugate(LogBarnesG(z)), Otherwise)))),
    Variables(z),
    Assumptions(Element(z, CC)))

5babc2

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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`BarnesG`	$G(z)$	Barnes G-function
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("5babc2"),
    Formula(Equal(ComplexDerivative(BarnesG(z), For(z, z)), Mul(BarnesG(z), Add(Sub(Mul(Sub(z, 1), DigammaFunction(z)), z), Div(Add(Log(Mul(2, Pi)), 1), 2))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

af31ae

\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}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

\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}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("af31ae"),
    Formula(Equal(ComplexBranchDerivative(Brackets(LogBarnesG(z)), For(z, z)), Add(Sub(Mul(Sub(z, 1), DigammaFunction(z)), z), Div(Add(Log(Mul(2, Pi)), 1), 2)))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

e05807

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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("e05807"),
    Formula(Equal(LogBarnesG(z), Add(Sub(Mul(Sub(z, 1), LogGamma(z)), HurwitzZeta(-1, z, 1)), ComplexDerivative(RiemannZeta(s), For(s, -1))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

0ad263

\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}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

TeX:

\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}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("0ad263"),
    Formula(Equal(LogBarnesG(Add(1, z)), Add(Sub(Mul(Div(Sub(Log(Mul(2, Pi)), 1), 2), z), Mul(Div(Add(1, ConstGamma), 2), Pow(z, 2))), Sum(Mul(Div(Mul(Pow(-1, Add(n, 1)), RiemannZeta(Sub(n, 1))), n), Pow(z, n)), For(n, 3, Infinity))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(z), 1))))

54d4e2

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]

Assumptions:

z \in \mathbb{C}

TeX:

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]

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Product`	$\prod_{n} f(n)$	Product
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("54d4e2"),
    Formula(Equal(BarnesG(Add(z, 1)), Mul(Mul(Pow(Mul(2, Pi), Div(z, 2)), Exp(Neg(Div(Add(z, Mul(Add(ConstGamma, 1), Pow(z, 2))), 2)))), Product(Brackets(Mul(Pow(Add(1, Div(z, k)), k), Exp(Sub(Div(Pow(z, 2), Mul(2, k)), z)))), For(k, 1, Infinity))))),
    Variables(z),
    Assumptions(Element(z, CC)))

752bde

Symbol: LogBarnesGRemainder —

R_{N}\!\left(z\right)

— Remainder term in asymptotic expansion of logarithmic Barnes G-function

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function

Source code for this entry:

Entry(ID("752bde"),
    SymbolDefinition(LogBarnesGRemainder, LogBarnesGRemainder(N, z), "Remainder term in asymptotic expansion of logarithmic Barnes G-function"))

6f8e14

\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)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

\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)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Pow`	${a}^{b}$	Power
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Log`	$\log(z)$	Natural logarithm
`Sum`	$\sum_{n} f(n)$	Sum
`BernoulliB`	$B_{n}$	Bernoulli number
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("6f8e14"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Add(Add(Sub(Sub(Add(Div(Pow(z, 2), 4), Mul(z, LogGamma(Add(z, 1)))), Mul(Add(Div(Mul(z, Add(z, 1)), 2), Div(1, 12)), Log(z))), Log(ConstGlaisher)), Sum(Div(BernoulliB(Add(Mul(2, n), 2)), Mul(Mul(Mul(Mul(2, n), Add(Mul(2, n), 1)), Add(Mul(2, n), 2)), Pow(z, Mul(2, n)))), For(n, 1, Sub(N, 1)))), LogBarnesGRemainder(N, z)))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), Element(N, ZZGreaterEqual(1)))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

b16d00

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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sum`	$\sum_{n} f(n)$	Sum
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("b16d00"),
    Formula(Equal(LogBarnesGRemainder(N, z), Integral(Mul(Sub(Mul(Div(t, 2), Coth(Div(t, 2))), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Pow(t, Mul(2, k))), For(k, 0, N))), Div(Exp(Neg(Mul(z, t))), Pow(t, 3))), For(t, 0, Infinity)))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(N, ZZGreaterEqual(1)))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

092cee

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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("092cee"),
    Formula(Equal(LogBarnesGRemainder(N, z), Mul(Mul(Div(1, Pow(z, Mul(2, N))), Div(Pow(-1, Add(N, 1)), Pi)), Integral(Mul(Div(Pow(t, Sub(Mul(2, N), 1)), Add(1, Pow(Div(t, z), 2))), PolyLog(2, Exp(Neg(Mul(Mul(2, Pi), t))))), For(t, 0, Infinity))))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(N, ZZGreaterEqual(1)))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

645c98

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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("645c98"),
    Formula(Equal(LogBarnesGRemainder(N, z), Mul(Div(1, Mul(Mul(2, N), Add(Mul(2, N), 1))), Integral(Div(BernoulliPolynomial(Add(Mul(2, N), 1), Sub(t, Floor(t))), Pow(Add(t, z), Mul(2, N))), For(t, 0, Infinity))))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(N, ZZGreaterEqual(1)))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

1d4638

\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}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

\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}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`Arg`	$\arg(z)$	Complex argument
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("1d4638"),
    Formula(LessEqual(Abs(LogBarnesGRemainder(N, z)), Mul(Div(Abs(BernoulliB(Add(Mul(2, N), 2))), Mul(Mul(Mul(Mul(2, N), Add(Mul(2, N), 1)), Add(Mul(2, N), 2)), Pow(Abs(z), Mul(2, N)))), Cases(Tuple(1, LessEqual(Abs(Arg(z)), Div(Pi, 4))), Tuple(Pow(Sec(Mul(Div(1, 2), Arg(z))), Add(Mul(2, N), 1)), Less(Abs(Arg(z)), Pi)))))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), Element(N, ZZGreaterEqual(1)))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

8c96a5

\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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right]

References:

https://arxiv.org/abs/math/0308086

TeX:

\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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right]

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("8c96a5"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Sub(Add(Add(Div(Mul(z, Sub(1, z)), 2), Mul(Div(z, 2), Log(Mul(2, Pi)))), Mul(z, LogGamma(z))), Integral(LogGamma(x), For(x, 0, z))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), -1)))),
    References("https://arxiv.org/abs/math/0308086"))

95f771

\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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right]

References:

https://arxiv.org/abs/math/0308086

TeX:

\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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -1\right]

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("95f771"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Add(Add(Div(Mul(z, Sub(1, z)), 2), Mul(Div(z, 2), Log(Mul(2, Pi)))), Integral(Mul(x, DigammaFunction(x)), For(x, 0, z))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), -1)))),
    References("https://arxiv.org/abs/math/0308086"))

b64782

\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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

References:

https://arxiv.org/abs/math/0308086

TeX:

\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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("b64782"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Sub(Sub(Add(Add(Mul(Div(Pow(z, 2), 4), Sub(Mul(2, Log(z)), 3)), Div(Mul(z, Log(Mul(2, Pi))), 2)), Div(1, 12)), Log(ConstGlaisher)), Integral(Div(Mul(x, Log(Add(Pow(x, 2), Pow(z, 2)))), Sub(Exp(Mul(Mul(2, Pi), x)), 1)), For(x, 0, Infinity))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0))),
    References("https://arxiv.org/abs/math/0308086"))

6395ee

\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

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

References:

https://arxiv.org/abs/math/0308086

TeX:

\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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("6395ee"),
    Formula(Equal(LogBarnesG(Add(z, 1)), Sub(Sub(Sub(Add(Mul(z, LogGamma(z)), Div(Pow(z, 2), 4)), Mul(Div(Log(z), 2), BernoulliPolynomial(2, z))), Log(ConstGlaisher)), Integral(Mul(Div(Exp(Mul(Neg(z), x)), Pow(x, 2)), Sub(Sub(Sub(Div(1, Sub(1, Exp(Neg(x)))), Div(1, x)), Div(1, 2)), Div(x, 12))), For(x, 0, Infinity))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0))),
    References("https://arxiv.org/abs/math/0308086"))

4a3612

\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)

Assumptions:

x \in \left(0, \infty\right)

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

\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)

x \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("4a3612"),
    Formula(Less(LogBarnesG(Add(x, 1)), Sub(Add(Add(Sub(Mul(Sub(Div(Pow(x, 2), 2), Div(1, 12)), Log(x)), Div(Mul(3, Pow(x, 2)), 4)), Mul(Div(Log(Mul(2, Pi)), 2), x)), Div(1, 12)), Log(ConstGlaisher)))),
    Variables(x),
    Assumptions(Element(x, OpenInterval(0, Infinity))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

3544a0

\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}}

Assumptions:

x \in \left(0, \infty\right)

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

\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}}

x \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("3544a0"),
    Formula(Greater(LogBarnesG(Add(x, 1)), Sub(Sub(Add(Add(Sub(Mul(Sub(Div(Pow(x, 2), 2), Div(1, 12)), Log(x)), Div(Mul(3, Pow(x, 2)), 4)), Mul(Div(Log(Mul(2, Pi)), 2), x)), Div(1, 12)), Log(ConstGlaisher)), Div(1, Mul(240, Pow(x, 2)))))),
    Variables(x),
    Assumptions(Element(x, OpenInterval(0, Infinity))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

7df1c4

G(x) > 0

Assumptions:

x \in \left(0, \infty\right)

TeX:

G(x) > 0

x \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("7df1c4"),
    Formula(Greater(BarnesG(x), 0)),
    Variables(x),
    Assumptions(Element(x, OpenInterval(0, Infinity))))

1c770c

\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}

Assumptions:

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \left\{0, 1, 2, 3\right\}

TeX:

\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}

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \left\{0, 1, 2, 3\right\}

Definitions:

Fungrim symbol	Notation	Short description
`RealDerivative`	$\frac{d}{d x}\, f\!\left(x\right)$	Real derivative
`BarnesG`	$G(z)$	Barnes G-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("1c770c"),
    Formula(Where(Implies(Greater(x, Subscript(x, 0)), Greater(RealDerivative(BarnesG(x), For(x, x, n)), 0)), Equal(Subscript(x, 0), Cases(Tuple(0, Equal(n, 0)), Tuple(Decimal("2.557664"), Equal(n, 1)), Tuple(Decimal("1.898850"), Equal(n, 2)), Tuple(Decimal("0.788740"), Equal(n, 3)))))),
    Variables(x, n),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(n, Set(0, 1, 2, 3)))))

306699

\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}

Assumptions:

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \left\{0, 1, 2, 3\right\}

TeX:

\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}

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \left\{0, 1, 2, 3\right\}

Definitions:

Fungrim symbol	Notation	Short description
`RealDerivative`	$\frac{d}{d x}\, f\!\left(x\right)$	Real derivative
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("306699"),
    Formula(Where(Implies(Greater(x, Subscript(x, 0)), Greater(RealDerivative(Brackets(LogBarnesG(x)), For(x, x, n)), 0)), Equal(Subscript(x, 0), Cases(Tuple(3, Equal(n, 0)), Tuple(Decimal("2.557664"), Equal(n, 1)), Tuple(Decimal("1.925864"), Equal(n, 2)), Tuple(0, Equal(n, 3)))))),
    Variables(x, n),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(n, Set(0, 1, 2, 3)))))

dc6806

\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)

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\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)

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`BellNumber`	$B_{n}$	Bell number
`Product`	$\prod_{n} f(n)$	Product
`Factorial`	$n !$	Factorial
`BarnesG`	$G(z)$	Barnes G-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("dc6806"),
    Formula(Equal(Det(Matrix(BellNumber(Add(i, j)), For(i, 0, n), For(j, 0, n))), Product(Factorial(k), For(k, 1, n)), BarnesG(Add(n, 2)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

f4750b

\operatorname{det}\!\left(H_{n}\right) = \frac{{\left(G\!\left(n + 1\right)\right)}^{4}}{G\!\left(2 n + 1\right)}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\operatorname{det}\!\left(H_{n}\right) = \frac{{\left(G\!\left(n + 1\right)\right)}^{4}}{G\!\left(2 n + 1\right)}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`BarnesG`	$G(z)$	Barnes G-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("f4750b"),
    Formula(Equal(Det(HilbertMatrix(n)), Div(Pow(BarnesG(Add(n, 1)), 4), BarnesG(Add(Mul(2, n), 1))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Definitions

Illustrations

Domain

Barnes G-function

Logarithmic Barnes G-function

Logarithmic form

Specific values

Integers

Rational arguments

Derivatives

Singularities and zeros

Zeros

Branch cuts

Functional equations

Recurrence relation

Reflection formula, real variables

Reflection formula, complex variables

Multiplication theorem

Conjugate symmetry

Derivatives and differential equations

Representation by other functions

Series and product representations

Taylor series

Weierstrass product

Asymptotic expansion

Integral representations

Bounds and inequalities

Upper and lower bounds

Monotonicity and convexity

Matrix formulas