# Fungrim entry: 4e4380

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

y \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \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
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
Abs$\left|z\right|$ Absolute value
Arg$\arg(z)$ Complex argument
Source code for this entry:
Entry(ID("4e4380"),
Formula(Equal(CarlsonRD(0, y, z), Mul(Pow(y, Neg(Div(3, 2))), Cases(Tuple(Div(Mul(3, Sub(EllipticE(Sub(1, Div(z, y))), Mul(Div(z, y), EllipticK(Sub(1, Div(z, y)))))), Mul(Div(z, y), Sub(1, Div(z, y)))), And(NotEqual(z, 0), NotEqual(z, y))), Tuple(Div(Mul(3, Pi), 4), Equal(z, y)), Tuple(UnsignedInfinity, Equal(z, 0)))))),
Variables(y, z),
Assumptions(And(Element(y, SetMinus(CC, Set(0))), Element(z, 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