# Fungrim entry: 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)))))))

## Topics using this entry

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