# 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

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