All the properties of $\operatorname{cis}$ operator using De Moivre's formula

Asked Sep 03 '21 at 19:13

Active Sep 05 '21 at 11:59

Viewed 91 times

I know that

$$\boxed{(\operatorname{cis}(\varphi))^n=(\cos \varphi +i \sin \varphi)^n\equiv \cos(n\varphi)+i\sin(n\varphi)} \tag 1$$

I would like to know if there are several properties of the $\operatorname{cis}$ operator using De Moivre's formula further this: $$\operatorname{cis}(\phi)^n=\operatorname{cis}(n\phi)$$

edited Sep 05 '21 at 11:59

DavidM.

asked Sep 03 '21 at 19:13

Sebastiano

7,649

1

See this answer of mine – FShrike Sep 03 '21 at 19:23
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It is essentially the same, just replace the roots with powers $\gt1$ – FShrike Sep 03 '21 at 19:24
@FShrike Thank you very much and I give you my thanks. – Sebastiano Sep 03 '21 at 19:28
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Like $\operatorname{cis}(x+y)=\operatorname{cis}(x)\operatorname{cis}(y)$? – J.G. Sep 05 '21 at 11:51
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I have corrected "proprieties" $\to$ properties. – Jean Marie Sep 05 '21 at 11:54
@JeanMarie Thank you very very much $\ddot \smile$. Sometimes my Italian language is go out here. – Sebastiano Sep 05 '21 at 11:56
@J.G. Hi :-) but is it like the $\log$ or $\exp$ properties? Thank you very much also for you. – Sebastiano Sep 05 '21 at 11:57
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@Sebastiano The property I mentioned is like $\exp$ (indeed, $\operatorname{cis}(x)=\exp{ix}$, but maybe that's a spoiler for what comes next). – J.G. Sep 05 '21 at 12:04
@J.G. It is true :-(....I not remembered. :-(((( However I am glad to receive a gradite answer. :-) – Sebastiano Sep 05 '21 at 12:06

All the properties of $\operatorname{cis}$ operator using De Moivre's formula

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