Fungrim entry: 61c002

R_D\!\left(0, 1, z\right) = \begin{cases} \frac{3 \left(E\!\left(1 - z\right) - z K\!\left(1 - z\right)\right)}{z \left(1 - z\right)}, & z \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 1\\\frac{3 \pi}{4}, & z = 1\\{\tilde \infty}, & z = 0\\ \end{cases}

Assumptions:

z \in \mathbb{C}

TeX:

R_D\!\left(0, 1, z\right) = \begin{cases} \frac{3 \left(E\!\left(1 - z\right) - z K\!\left(1 - z\right)\right)}{z \left(1 - z\right)}, & z \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 1\\\frac{3 \pi}{4}, & z = 1\\{\tilde \infty}, & z = 0\\ \end{cases}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRD`	$R_D\!\left(x, y, z\right)$	Degenerate Carlson symmetric elliptic integral of the third kind
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("61c002"),
    Formula(Equal(CarlsonRD(0, 1, z), Cases(Tuple(Div(Mul(3, Sub(EllipticE(Sub(1, z)), Mul(z, EllipticK(Sub(1, z))))), Mul(z, Sub(1, z))), And(NotEqual(z, 0), NotEqual(z, 1))), Tuple(Div(Mul(3, Pi), 4), Equal(z, 1)), Tuple(UnsignedInfinity, Equal(z, 0))))),
    Variables(z),
    Assumptions(Element(z, CC)))

Topics using this entry

Specific values of Carlson symmetric elliptic integrals