# Powers

Symbol: Pow ${a}^{b}$ Power

## Integer exponents

${0}^{0} = 1$
${z}^{0} = 1$
${z}^{1} = z$
${z}^{n + 1} = {z}^{n} z$
${z}^{-1} = \frac{1}{z}$

## Elementary functions

${a}^{b} = \exp\!\left(b \log\!\left(a\right)\right)$
${z}^{1 / 2} = \sqrt{z}$
${z}^{-1 / 2} = \frac{1}{\sqrt{z}}$

## Complex parts

${\left(a + b i\right)}^{c + d i} = {M}^{c} {e}^{-d \theta} \left(\cos\!\left(c \theta + d \log\!\left(M\right)\right) + i \sin\!\left(c \theta + d \log\!\left(M\right)\right)\right)\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)$
$\left|{\left(a + b i\right)}^{c + d i}\right| = {M}^{c} {e}^{-d \theta}\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)$
$\operatorname{Re}\!\left({\left(a + b i\right)}^{c + d i}\right) = {M}^{c} {e}^{-d \theta} \cos\!\left(c \theta + d \log\!\left(M\right)\right)\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)$
$\operatorname{Im}\!\left({\left(a + b i\right)}^{c + d i}\right) = {M}^{c} {e}^{-d \theta} \sin\!\left(c \theta + d \log\!\left(M\right)\right)\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)$

## Expansion

${\left(x y\right)}^{a} = {x}^{a} {y}^{a} \exp\!\left(2 \pi i a \left\lfloor \frac{\pi - \arg\!\left(x\right) - \arg\!\left(y\right)}{2 \pi} \right\rfloor\right)$

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