# Complex parts

Symbol: Sign $\operatorname{sgn}\!\left(z\right)$ Sign function
Symbol: Abs $\left|z\right|$ Absolute value
Symbol: Arg $\arg\!\left(z\right)$ Complex argument
Symbol: Re $\operatorname{Re}\!\left(z\right)$ Real part
Symbol: Im $\operatorname{Im}\!\left(z\right)$ Imaginary part
Symbol: Conjugate $\overline{z}$ Complex conjugate

## Basic formulas

$\operatorname{sgn}\!\left(z\right) = \frac{z}{\left|z\right|}$
$\left|x + y i\right| = \sqrt{{x}^{2} + {y}^{2}}$
$\arg\!\left(x + y i\right) = \operatorname{atan2}\!\left(y, x\right)$
$\operatorname{Re}\!\left(x + y i\right) = x$
$\operatorname{Im}\!\left(x + y i\right) = y$
$\overline{x + y i} = x - y i$

## Specific values

$\arg\!\left(1\right) = 0$
$\arg\!\left(i\right) = \frac{\pi}{2}$
$\arg\!\left(-i\right) = -\frac{\pi}{2}$
$\arg\!\left(-1\right) = \pi$

## Connection formulas

$\operatorname{Re}\!\left(z\right) = \frac{z + \overline{z}}{2}$
$\operatorname{Im}\!\left(z\right) = \frac{z - \overline{z}}{2 i}$
$\operatorname{sgn}\!\left(z\right) = \exp\!\left(i \arg\!\left(z\right)\right)$
$\arg\!\left(z\right) = -i \log\!\left(\operatorname{sgn}\!\left(z\right)\right)$

## Functional equations

$z \overline{z} = {\left|z\right|}^{2}$
$\arg\!\left(c z\right) = \arg\!\left(z\right)$

## Bounds and inequalities

$\left|a b\right| = \left|a\right| \left|b\right|$
$\left|a + b\right| \le \left|a\right| + \left|b\right|$
$\left|\left|a\right| - \left|b\right|\right| \le \left|a - b\right|$
$\left|\overline{z}\right| = \left|z\right|$
$\left|\operatorname{Re}\!\left(z\right)\right| \le \left|z\right|$
$\left|\operatorname{Im}\!\left(z\right)\right| \le \left|z\right|$
$\left|z\right| \le \left|\operatorname{Re}\!\left(z\right)\right| + \left|\operatorname{Im}\!\left(z\right)\right|$
$\left|\operatorname{sgn}\!\left(z\right)\right| \le 1$
$\left|\arg\!\left(z\right)\right| \le \pi$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC