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Fungrim entry: 0d8e03

ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
Assumptions:aC  and  b(0,)a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)
Alternative assumptions:a[0,)  and  bC(,0]a \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \left(-\infty, 0\right]
Alternative assumptions:aC  and  bC{0}  and  arg(a)arg(b)(π,π]a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \arg(a) - \arg(b) \in \left(-\pi, \pi\right]
TeX:
\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)

a \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \left(-\infty, 0\right]

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \arg(a) - \arg(b) \in \left(-\pi, \pi\right]
Definitions:
Fungrim symbol Notation Short description
Sqrtz\sqrt{z} Principal square root
CCC\mathbb{C} Complex numbers
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Argarg(z)\arg(z) Complex argument
Piπ\pi The constant pi (3.14...)
Source code for this entry:
Entry(ID("0d8e03"),
    Formula(Equal(Sqrt(Div(a, b)), Div(Sqrt(a), Sqrt(b)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, OpenInterval(0, Infinity))), And(Element(a, ClosedOpenInterval(0, Infinity)), Element(b, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))), And(Element(a, CC), Element(b, SetMinus(CC, Set(0))), Element(Sub(Arg(a), Arg(b)), OpenClosedInterval(Neg(Pi), Pi)))))

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