# Fungrim entry: 8d0629

$R_D\!\left(0, y, z\right) = {z}^{-3 / 2} \begin{cases} \frac{3 \left(K\!\left(1 - \frac{y}{z}\right) - E\!\left(1 - \frac{y}{z}\right)\right)}{1 - \frac{y}{z}}, & y \ne z\\\frac{3 \pi}{4}, & y = z\\ \end{cases}$
Assumptions:$z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(y) - \arg(z)\right| < \pi$
TeX:
R_D\!\left(0, y, z\right) = {z}^{-3 / 2} \begin{cases} \frac{3 \left(K\!\left(1 - \frac{y}{z}\right) - E\!\left(1 - \frac{y}{z}\right)\right)}{1 - \frac{y}{z}}, & y \ne z\\\frac{3 \pi}{4}, & y = z\\ \end{cases}

z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|\arg(y) - \arg(z)\right| < \pi
Definitions:
Fungrim symbol Notation Short description
CarlsonRD$R_D\!\left(x, y, z\right)$ Degenerate Carlson symmetric elliptic integral of the third kind
Pow${a}^{b}$ Power
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
Abs$\left|z\right|$ Absolute value
Arg$\arg(z)$ Complex argument
Source code for this entry:
Entry(ID("8d0629"),
Formula(Equal(CarlsonRD(0, y, z), Mul(Pow(z, Neg(Div(3, 2))), Cases(Tuple(Div(Mul(3, Sub(EllipticK(Sub(1, Div(y, z))), EllipticE(Sub(1, Div(y, z))))), Sub(1, Div(y, z))), NotEqual(y, z)), Tuple(Div(Mul(3, Pi), 4), Equal(y, z)))))),
Variables(y, z),
Assumptions(And(Element(z, SetMinus(CC, Set(0))), Element(y, CC), Less(Abs(Sub(Arg(y), Arg(z))), Pi))))

## 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