# Fungrim entry: 3e1435

$\frac{d}{d x}\, R_G\!\left(x, y, z\right) + \frac{d}{d y}\, R_G\!\left(x, y, z\right) + \frac{d}{d z}\, R_G\!\left(x, y, z\right) = \frac{1}{2} R_F\!\left(x, y, z\right)$
Assumptions:$x \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right]$
TeX:
x \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right]
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
CarlsonRG$R_G\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the second kind
CarlsonRF$R_F\!\left(x, y, z\right)$ Carlson symmetric elliptic integral of the first kind
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("3e1435"),
Equal(Add(Add(ComplexDerivative(CarlsonRG(x, y, z), For(x, x)), ComplexDerivative(CarlsonRG(x, y, z), For(y, y))), ComplexDerivative(CarlsonRG(x, y, z), For(z, z))), Mul(Div(1, 2), CarlsonRF(x, y, z))),
Variables(x, y, z),
Assumptions(And(Element(x, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))))

## 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