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Halphen's constant

Table of contents: Definitions - Numerical value - Approximation theory - Formulas

Definitions

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Symbol: HalphenConstant Λ\Lambda Halphen's constant (one-ninth constant) 0.10765...

Numerical value

e2bfdb
Λ=0.10765391922648457661532344509094719058797656329012  (nearest 50 digits)\Lambda = 0.10765391922648457661532344509094719058797656329012 \;\, {\scriptstyle (\text{nearest } 50 \text{ digits})}
f5e0b0
1Λ=9.2890254919208189187554494359517450610316948677501  (nearest 50 digits)\frac{1}{\Lambda} = 9.2890254919208189187554494359517450610316948677501 \;\, {\scriptstyle (\text{nearest } 50 \text{ digits})}
d0993b
Λ19\Lambda \ne \frac{1}{9}

Approximation theory

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Λ=limnλn1/n   where R={r:rR(t)anddeg(r)(n,n)},  λn=infrRsupx(,0]exr(x)\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|

Formulas

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Λ=zero*x(0,1)(18+n=1nxn1(x)n)\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)
31adf6
Λ=zero*x(0,1)[n=0(2n+1)2(x)n(n+1)/2]\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]
831ea4
Λ=zero*x(0,1)(18+n=1dn(1)ddxn)\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)
c26bc9
Λ=exp ⁣(πK ⁣(1c)K(c))   where c=zero*m(0,1)(K(m)2E(m))\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)
06c468
Λ=zero*x(0,1)[θ2 ⁣(0,log ⁣(x)2πi)]\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]

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

2020-01-31 18:09:28.494564 UTC