# Fungrim entry: 3e05c6

$R_D\!\left(0, y, 1\right) = \begin{cases} \frac{3 \left(K\!\left(1 - y\right) - E\!\left(1 - y\right)\right)}{1 - y}, & y \ne 1\\\frac{3 \pi}{4}, & y = 1\\ \end{cases}$
Assumptions:$y \in \mathbb{C}$
TeX:
R_D\!\left(0, y, 1\right) = \begin{cases} \frac{3 \left(K\!\left(1 - y\right) - E\!\left(1 - y\right)\right)}{1 - y}, & y \ne 1\\\frac{3 \pi}{4}, & y = 1\\ \end{cases}

y \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
EllipticK$K(m)$ Legendre complete elliptic integral of the first kind
EllipticE$E(m)$ Legendre complete elliptic integral of the second kind
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("3e05c6"),
Formula(Equal(CarlsonRD(0, y, 1), Cases(Tuple(Div(Mul(3, Sub(EllipticK(Sub(1, y)), EllipticE(Sub(1, y)))), Sub(1, y)), NotEqual(y, 1)), Tuple(Div(Mul(3, Pi), 4), Equal(y, 1))))),
Variables(y),
Assumptions(Element(y, 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