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"),
    Equal(ComplexDerivative(CarlsonRG(Add(x, t), Add(y, t), Add(z, t)), For(t, t)), Mul(Div(1, 2), CarlsonRF(Add(x, t), Add(y, t), Add(z, t)))),
    Variables(x, y, z, t),
    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)))))

Topics using this entry

Carlson symmetric elliptic integrals