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Fungrim entry: 114f9e

f(α) is holomorphic on αC(,0]   for all f{αRD ⁣(α,y,z),αRD ⁣(x,α,z),αRD ⁣(x,y,α)}f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_D\!\left(\alpha, y, z\right), \alpha \mapsto R_D\!\left(x, \alpha, z\right), \alpha \mapsto R_D\!\left(x, y, \alpha\right)\right\}
Assumptions:xC  and  yC  and  zC{0}  and  (x0  or  y0)x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)
TeX:
f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_D\!\left(\alpha, y, z\right), \alpha \mapsto R_D\!\left(x, \alpha, z\right), \alpha \mapsto R_D\!\left(x, y, \alpha\right)\right\}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)
Definitions:
Fungrim symbol Notation Short description
IsHolomorphicf(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
CarlsonRDRD ⁣(x,y,z)R_D\!\left(x, y, z\right) Degenerate Carlson symmetric elliptic integral of the third kind
Source code for this entry:
Entry(ID("114f9e"),
    Formula(All(IsHolomorphic(f(alpha), ForElement(alpha, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))), ForElement(f, Set(Fun(alpha, CarlsonRD(alpha, y, z)), Fun(alpha, CarlsonRD(x, alpha, z)), Fun(alpha, CarlsonRD(x, y, alpha)))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, SetMinus(CC, Set(0))), Or(NotEqual(x, 0), NotEqual(y, 0)))))

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2020-04-08 16:14:44.404316 UTC