Fungrim entry: 741859

\frac{d}{d y}\, R_C\!\left(x, y\right) = \begin{cases} \frac{1}{2 \left(x - y\right)} \left(R_C\!\left(x, y\right) - \frac{\sqrt{x}}{y}\right), & x \ne y\\-\frac{1}{3} {x}^{-3 / 2}, & x = y\\ \end{cases}

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}}\; x \ne y

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}}\; x \ne y

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("741859"),
    Equal(ComplexDerivative(CarlsonRC(x, y), For(y, y)), Cases(Tuple(Mul(Div(1, Mul(2, Sub(x, y))), Sub(CarlsonRC(x, y), Div(Sqrt(x), y))), NotEqual(x, y)), Tuple(Neg(Mul(Div(1, 3), Pow(x, Neg(Div(3, 2))))), Equal(x, y)))),
    Variables(x, y),
    Assumptions(And(Element(x, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), NotEqual(x, y))))

Topics using this entry

Carlson symmetric elliptic integrals