# 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

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