Fungrim entry: e722ca

\operatorname{Im}\!\left(\sqrt{z}\right) = \operatorname{sgn}\!\left(\operatorname{Im}(z)\right) \sqrt{\frac{\left|z\right| - \operatorname{Re}(z)}{2}}

Assumptions:

z \in \mathbb{C} \setminus \left(-\infty, 0\right)

TeX:

\operatorname{Im}\!\left(\sqrt{z}\right) = \operatorname{sgn}\!\left(\operatorname{Im}(z)\right) \sqrt{\frac{\left|z\right| - \operatorname{Re}(z)}{2}}

z \in \mathbb{C} \setminus \left(-\infty, 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Sqrt`	$\sqrt{z}$	Principal square root
`Sign`	$\operatorname{sgn}(z)$	Sign function
`Abs`	$\left\|z\right\|$	Absolute value
`Re`	$\operatorname{Re}(z)$	Real part
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("e722ca"),
    Formula(Equal(Im(Sqrt(z)), Mul(Sign(Im(z)), Sqrt(Div(Sub(Abs(z), Re(z)), 2))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, OpenInterval(Neg(Infinity), 0)))))

Topics using this entry

Square roots