My Coding > Mathematics > Complex numbers > Cubic root from (1+i)

Cubic root from (1+i)

Let's calculate cubic root from (1+i), or in another style of writing: ∛(1+i). This is very easy to do in Euler or the exponential form of complex numbers.

How to do it?

\[ \begin{align*} z &= 1 + i \\ |z| &= \sqrt{1^2 + 1^2} = \sqrt{2} \\ \theta &= \tan^{-1}\frac{1}{1} = \frac{\pi}{4} \\ z &= \sqrt{2},e^{i\pi/4} \\ z^{1/3} &= \sqrt[3]{\sqrt{2}}e^{i(\pi/12 + 2k\pi/3)},; k=0,1,2 \\ &= \sqrt[6]{2},(\cos(\frac{\pi}{12} + \frac{2k\pi}{3}) + i\sin(\frac{\pi}{12} + \frac{2k\pi}{3})),; k=0,1,2 \end{align*}\]

For more details and some comments please check this video:


Published: 2023-05-08 23:40:02
Updated: 2023-05-17 21:35:59

Last 10 artitles


9 popular artitles
© 2020 MyCoding.uk -My blog about coding and further learning. This blog was writen with pure Perl and front-end output was performed with TemplateToolkit.