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

Halphen's constant