Cis

Cis notation is a polar notation for complex numbers. For all complex numbers $z$ , we can write $z=r\mathrm{cis }(\theta)=r\cos \theta + ir\sin \theta$ . Notice that $\mathrm{cis}$ is made up by the first letter of $\cos$ , $i$ , and the first letter of $\sin$ .

Once one gets used to the notation, it is almost always preferred to write $re^{i\theta}$ rather than $r\mathrm{cis }(\theta)$ , as Euler's formula states that

$e^{i\theta} = \cos \theta + i \sin \theta.$

This is so that one can more naturally use the properties of the complex exponential. One important example is De Moivre's Theorem, which states that

$\mathrm{cis}(r\theta) = (\mathrm{cis}(\theta))^r.$

This is more easily understood in the complex exponential form:

$e^{i(r\theta)} = (e^{i\theta})^r.$

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Cis

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