# Fungrim entry: 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))))))))

## 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