Halphen's constant

Symbol: HalphenConstant —

\Lambda

— Halphen's constant (one-ninth constant) 0.10765...

►HalphenConstant —

\Lambda

— Represents Halphen's constant, also known as the one-ninth constant.

►Div(1, HalphenConstant) —

\frac{1}{\Lambda}

— Represents the reciprocal of Halphen's constant, also called Varga's constant.

References:

S. Finch (2003), Mathematical Constants, Cambridge University Press, section 4.5
https://www.chebfun.org/examples/approx/Halphen.html
http://mathworld.wolfram.com/One-NinthConstant.html

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...

Source code for this entry:

Entry(ID("6161c7"),
    SymbolDefinition(HalphenConstant, HalphenConstant, "Halphen's constant (one-ninth constant) 0.10765..."),
    CodeExample(HalphenConstant, "Represents Halphen's constant, also known as the one-ninth constant."),
    CodeExample(Div(1, HalphenConstant), "Represents the reciprocal of Halphen's constant, also called Varga's constant."),
    References("S. Finch (2003), Mathematical Constants, Cambridge University Press, section 4.5", "https://www.chebfun.org/examples/approx/Halphen.html", "http://mathworld.wolfram.com/One-NinthConstant.html"))

e2bfdb

\Lambda = 0.10765391922648457661532344509094719058797656329012 \;\, {\scriptstyle (\text{nearest } 50 \text{ digits})}

References:

Sequence A072558 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

\Lambda = 0.10765391922648457661532344509094719058797656329012 \;\, {\scriptstyle (\text{nearest } 50 \text{ digits})}

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS

Source code for this entry:

Entry(ID("e2bfdb"),
    Formula(EqualNearestDecimal(HalphenConstant, Decimal("0.10765391922648457661532344509094719058797656329012"), 50)),
    References(SloaneA("A072558")))

f5e0b0

\frac{1}{\Lambda} = 9.2890254919208189187554494359517450610316948677501 \;\, {\scriptstyle (\text{nearest } 50 \text{ digits})}

References:

Sequence A073007 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

\frac{1}{\Lambda} = 9.2890254919208189187554494359517450610316948677501 \;\, {\scriptstyle (\text{nearest } 50 \text{ digits})}

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS

Source code for this entry:

Entry(ID("f5e0b0"),
    Formula(EqualNearestDecimal(Div(1, HalphenConstant), Decimal("9.2890254919208189187554494359517450610316948677501"), 50)),
    References(SloaneA("A073007")))

d0993b

\Lambda \ne \frac{1}{9}

TeX:

\Lambda \ne \frac{1}{9}

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...

Source code for this entry:

Entry(ID("d0993b"),
    Formula(NotEqual(HalphenConstant, Div(1, 9))))

5c1e44

\Lambda = \lim_{n \to \infty} {\lambda}_{n}^{1 / n}\; \text{ where } R = \left\{ r : r \in \mathbb{R}(t) \,\mathbin{\operatorname{and}}\, \deg(r) \le \left(n, n\right) \right\},\;{\lambda}_{n} = \mathop{\operatorname{inf}}\limits_{r \in R} \mathop{\operatorname{sup}}\limits_{x \in \left(-\infty, 0\right]} \left|{e}^{x} - r(x)\right|

TeX:

\Lambda = \lim_{n \to \infty} {\lambda}_{n}^{1 / n}\; \text{ where } R = \left\{ r : r \in \mathbb{R}(t) \,\mathbin{\operatorname{and}}\, \deg(r) \le \left(n, n\right) \right\},\;{\lambda}_{n} = \mathop{\operatorname{inf}}\limits_{r \in R} \mathop{\operatorname{sup}}\limits_{x \in \left(-\infty, 0\right]} \left|{e}^{x} - r(x)\right|

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers
`Infimum`	$\mathop{\operatorname{inf}}\limits_{x \in S} f(x)$	Infimum of a set or function
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	Supremum of a set or function
`Abs`	$\left\|z\right\|$	Absolute value
`Exp`	${e}^{z}$	Exponential function
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval

Source code for this entry:

Entry(ID("5c1e44"),
    Formula(Equal(HalphenConstant, Where(SequenceLimit(Pow(Subscript(lamda, n), Div(1, n)), For(n, Infinity)), Equal(R, Set(r, ForElement(r, RationalFunctions(RR, t)), LessEqual(RationalFunctionDegree(r), Tuple(n, n)))), Equal(Subscript(lamda, n), Infimum(Supremum(Abs(Sub(Exp(x), r(x))), ForElement(x, OpenClosedInterval(Neg(Infinity), 0))), ForElement(r, R)))))))

9758ac

\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[-\frac{1}{8} + \sum_{n=1}^{\infty} \frac{n {x}^{n}}{1 - {\left(-x\right)}^{n}}\right]

TeX:

\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[-\frac{1}{8} + \sum_{n=1}^{\infty} \frac{n {x}^{n}}{1 - {\left(-x\right)}^{n}}\right]

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("9758ac"),
    Formula(Equal(HalphenConstant, UniqueZero(Add(Neg(Div(1, 8)), Sum(Div(Mul(n, Pow(x, n)), Sub(1, Pow(Neg(x), n))), For(n, 1, Infinity))), ForElement(x, OpenInterval(0, 1))))))

31adf6

\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[\sum_{n=0}^{\infty} {\left(2 n + 1\right)}^{2} {\left(-x\right)}^{n \left(n + 1\right) / 2}\right]

TeX:

\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[\sum_{n=0}^{\infty} {\left(2 n + 1\right)}^{2} {\left(-x\right)}^{n \left(n + 1\right) / 2}\right]

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("31adf6"),
    Formula(Equal(HalphenConstant, UniqueZero(Brackets(Sum(Mul(Pow(Add(Mul(2, n), 1), 2), Pow(Neg(x), Div(Mul(n, Add(n, 1)), 2))), For(n, 0, Infinity))), ForElement(x, OpenInterval(0, 1))))))

831ea4

\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[-\frac{1}{8} + \sum_{n=1}^{\infty} \left|\sum_{d \mid n} {\left(-1\right)}^{d} d\right| {x}^{n}\right]

TeX:

\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[-\frac{1}{8} + \sum_{n=1}^{\infty} \left|\sum_{d \mid n} {\left(-1\right)}^{d} d\right| {x}^{n}\right]

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`Sum`	$\sum_{n} f(n)$	Sum
`Abs`	$\left\|z\right\|$	Absolute value
`DivisorSum`	$\sum_{k \mid n} f(k)$	Sum over divisors
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("831ea4"),
    Formula(Equal(HalphenConstant, UniqueZero(Add(Neg(Div(1, 8)), Sum(Mul(Abs(DivisorSum(Mul(Pow(-1, d), d), For(d, n))), Pow(x, n)), For(n, 1, Infinity))), ForElement(x, OpenInterval(0, 1))))))

c26bc9

\Lambda = \exp\!\left(-\frac{\pi K\!\left(1 - c\right)}{K(c)}\right)\; \text{ where } c = \mathop{\operatorname{zero*}\,}\limits_{m \in \left(0, 1\right)} \left[K(m) - 2 E(m)\right]

TeX:

\Lambda = \exp\!\left(-\frac{\pi K\!\left(1 - c\right)}{K(c)}\right)\; \text{ where } c = \mathop{\operatorname{zero*}\,}\limits_{m \in \left(0, 1\right)} \left[K(m) - 2 E(m)\right]

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("c26bc9"),
    Formula(Equal(HalphenConstant, Where(Exp(Neg(Div(Mul(Pi, EllipticK(Sub(1, c))), EllipticK(c)))), Equal(c, UniqueZero(Sub(EllipticK(m), Mul(2, EllipticE(m))), ForElement(m, OpenInterval(0, 1))))))))

06c468

\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[\theta''_{2}\!\left(0 , \frac{\log\!\left(-x\right)}{2 \pi i}\right)\right]

TeX:

\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[\theta''_{2}\!\left(0 , \frac{\log\!\left(-x\right)}{2 \pi i}\right)\right]

Definitions:

Fungrim symbol	Notation	Short description
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("06c468"),
    Formula(Equal(HalphenConstant, UniqueZero(Brackets(JacobiTheta(2, 0, Div(Log(Neg(x)), Mul(Mul(2, Pi), ConstI)), 2)), ForElement(x, OpenInterval(0, 1))))))

Definitions

Numerical value

Approximation theory

Formulas