Fungrim entry: 987e3c

\operatorname{Im}(z) \in \left(-\pi, \pi\right] \;\implies\; \left(\log\!\left({e}^{z}\right) = z\right)

Assumptions:

z \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
`Im`	$\operatorname{Im}(z)$	Imaginary part
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Pi`	$\pi$	The constant pi (3.14...)
`Log`	$\log(z)$	Natural logarithm
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	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

Exponential function

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