# Fungrim entry: 114f9e

$f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_D\!\left(\alpha, y, z\right), \alpha \mapsto R_D\!\left(x, \alpha, z\right), \alpha \mapsto R_D\!\left(x, y, \alpha\right)\right\}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)$
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
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
CarlsonRD$R_D\!\left(x, y, z\right)$ 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)))))

## Topics using this entry

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