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Fungrim entry: c56825

f(α) is holomorphic on αC(,0]   for all f{αRG ⁣(α,y,z),αRG ⁣(x,α,z),αRG ⁣(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_G\!\left(\alpha, y, z\right), \alpha \mapsto R_G\!\left(x, \alpha, z\right), \alpha \mapsto R_G\!\left(x, y, \alpha\right)\right\}
Assumptions:xC  and  yC  and  zCx \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
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_G\!\left(\alpha, y, z\right), \alpha \mapsto R_G\!\left(x, \alpha, z\right), \alpha \mapsto R_G\!\left(x, y, \alpha\right)\right\}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
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
CarlsonRGRG ⁣(x,y,z)R_G\!\left(x, y, z\right) Carlson symmetric elliptic integral of the second kind
Source code for this entry:
Entry(ID("c56825"),
    Formula(All(IsHolomorphic(f(alpha), ForElement(alpha, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))), ForElement(f, Set(Fun(alpha, CarlsonRG(alpha, y, z)), Fun(alpha, CarlsonRG(x, alpha, z)), Fun(alpha, CarlsonRG(x, y, alpha)))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC