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

Halphen's constant