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Fungrim entry: 987e3c

Im(z)(π,π]        (log ⁣(ez)=z)\operatorname{Im}(z) \in \left(-\pi, \pi\right] \;\implies\; \left(\log\!\left({e}^{z}\right) = z\right)
Assumptions:zCz \in \mathbb{C}
TeX:
\operatorname{Im}(z) \in \left(-\pi, \pi\right] \;\implies\; \left(\log\!\left({e}^{z}\right) = z\right)

z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
ImIm(z)\operatorname{Im}(z) Imaginary part
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Piπ\pi The constant pi (3.14...)
Loglog(z)\log(z) Natural logarithm
Expez{e}^{z} Exponential function
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("987e3c"),
    Formula(Implies(Element(Im(z), OpenClosedInterval(Neg(Pi), Pi)), Equal(Log(Exp(z)), z))),
    Variables(z),
    Assumptions(Element(z, CC)))

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