Fungrim entry: b8ca70

R_J\!\left(x, y, z, w\right) = \lim_{\varepsilon \to {0}^{+}} R_J\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i, w + \varepsilon i\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)

TeX:

R_J\!\left(x, y, z, w\right) = \lim_{\varepsilon \to {0}^{+}} R_J\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i, w + \varepsilon i\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRJ`	$R_J\!\left(x, y, z, w\right)$	Carlson symmetric elliptic integral of the third kind
`RightLimit`	$\lim_{x \to {a}^{+}} f(x)$	Limiting value, from the right
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b8ca70"),
    Formula(Equal(CarlsonRJ(x, y, z, w), RightLimit(CarlsonRJ(Add(x, Mul(epsilon, ConstI)), Add(y, Mul(epsilon, ConstI)), Add(z, Mul(epsilon, ConstI)), Add(w, Mul(epsilon, ConstI))), For(epsilon, 0)))),
    Variables(x, y, z, w),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(w, SetMinus(CC, Set(0))), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))

Topics using this entry

Carlson symmetric elliptic integrals