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Fungrim entry: e722ca

Im ⁣(z)=sgn ⁣(Im(z))zRe(z)2\operatorname{Im}\!\left(\sqrt{z}\right) = \operatorname{sgn}\!\left(\operatorname{Im}(z)\right) \sqrt{\frac{\left|z\right| - \operatorname{Re}(z)}{2}}
Assumptions:zC(,0)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
ImIm(z)\operatorname{Im}(z) Imaginary part
Sqrtz\sqrt{z} Principal square root
Signsgn(z)\operatorname{sgn}(z) Sign function
Absz\left|z\right| Absolute value
ReRe(z)\operatorname{Re}(z) Real part
CCC\mathbb{C} Complex numbers
OpenInterval(a,b)\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