Fungrim entry: 0d8e03

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

Assumptions:

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

Alternative assumptions:

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

Alternative assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \arg\!\left(a\right) - \arg\!\left(b\right) \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\!\left(a\right) - \arg\!\left(b\right) \in \left(-\pi, \pi\right]

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Arg`	$\arg\!\left(z\right)$	Complex argument
`ConstPi`	$\pi$	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(ConstPi), ConstPi)))))

Topics using this entry

Square roots