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Fungrim entry: 5c1e44

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|
\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|
Fungrim symbol Notation Short description
HalphenConstantΛ\Lambda Halphen's constant (one-ninth constant) 0.10765...
SequenceLimitlimnaf(n)\lim_{n \to a} f(n) Limiting value of sequence
Powab{a}^{b} Power
Infinity\infty Positive infinity
RRR\mathbb{R} Real numbers
InfimuminfxSf(x)\mathop{\operatorname{inf}}\limits_{x \in S} f(x) Infimum of a set or function
SupremumsupxSf(x)\mathop{\operatorname{sup}}\limits_{x \in S} f(x) Supremum of a set or function
Absz\left|z\right| Absolute value
Expez{e}^{z} Exponential function
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Source code for this entry:
    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)))))))

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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