# Fungrim entry: c56825

$f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_G\!\left(\alpha, y, z\right), \alpha \mapsto R_G\!\left(x, \alpha, z\right), \alpha \mapsto R_G\!\left(x, y, \alpha\right)\right\}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$
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
IsHolomorphic$f(z) \text{ is holomorphic at } z = c$ Holomorphic predicate
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
CarlsonRG$R_G\!\left(x, y, z\right)$ 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))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC