# Imaginary unit

## Definitions

Symbol: ConstI $i$ Imaginary unit

## Domain

$i \in \mathbb{C}$
$i \in \overline{\mathbb{Q}}$
$i \notin \mathbb{R}$

$\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[{x}^{2} + 1 = 0\right] = \left\{i, -i\right\}$

## Numerical value

$i = \sqrt{-1}$
$i = {\left(-1\right)}^{1 / 2}$

## Complex parts

$\left|i\right| = 1$
$\operatorname{Re}(i) = 0$
$\operatorname{Im}(i) = 1$
$\arg(i) = \frac{\pi}{2}$
$\arg\!\left(-i\right) = -\frac{\pi}{2}$
$\operatorname{sgn}(i) = i$

## Transformations

${i}^{2} = -1$
${i}^{3} = -i$
${i}^{4} = 1$
${i}^{n} = \begin{cases} 1, & n \equiv 0 \pmod {4}\\i, & n \equiv 1 \pmod {4}\\-1, & n \equiv 2 \pmod {4}\\-i, & n \equiv 3 \pmod {4}\\ \end{cases}$
$\overline{i} = -i$
$\frac{1}{i} = -i$
${i}^{z} = {e}^{\pi i z / 2}$
${i}^{z} = \cos\!\left(\frac{\pi}{2} z\right) + \sin\!\left(\frac{\pi}{2} z\right) i$
$\sqrt{i} = \frac{1}{\sqrt{2}} \left(1 + i\right)$
${i}^{i} = {e}^{-\pi / 2}$

## Special functions at this value

$\log(i) = \frac{\pi i}{2}$
$\left|\Gamma(i)\right| = \sqrt{\frac{\pi}{\sinh(\pi)}}$
$\operatorname{Im}\!\left(\psi\!\left(i\right)\right) = \frac{1}{2} \left(\pi \coth(\pi) + 1\right)$
$\operatorname{Li}_{2}\!\left(i\right) = -\frac{{\pi}^{2}}{48} + G i$

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