# Fungrim entry: 638fa6

$\frac{d}{d t}\, R_G\!\left(x + t, y + t, z + t\right) = \frac{1}{2} R_F\!\left(x + t, y + t, z + t\right)$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; t \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x + t \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; y + t \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z + t \notin \left(-\infty, 0\right]$
TeX:
x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; t \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x + t \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; y + t \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; z + t \notin \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("638fa6"),
Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(t, CC), NotElement(Add(x, t), OpenClosedInterval(Neg(Infinity), 0)), NotElement(Add(y, t), OpenClosedInterval(Neg(Infinity), 0)), NotElement(Add(z, t), OpenClosedInterval(Neg(Infinity), 0)))))