Difference between revisions of "Cis"

Revision as of 17:33, 7 July 2006

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''' notation is a [[polar form | polar]] notation for [[complex number]]s. For all complex numbers <math>z</math>, we can write <math>z=r\mathrm{cis }(\theta)=r\cos \theta + ir\sin \theta</math>. Notice that <math>\mathrm{cis}</math> is made up by the first letter of <math>\cos</math>, <math>i</math>, and the first letter of <math>\sin</math>.
+Once one gets used to the notation, it is almost always preferred to write <math>re^{i\theta}</math> rather than <math>r\mathrm{cis }(\theta)</math>, as [[Euler's formula]] states that
+<math>e^{i\theta} = \cos \theta + i \sin \theta.</math>
+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
+<math>\mathrm{cis}(r\theta) = (\mathrm{cis}(\theta))^r.</math>
+This is more easily understood in the complex exponential form:
+<math>e^{i(r\theta)} = (e^{i\theta})^r.</math>
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Difference between revisions of "Cis"

Revision as of 17:33, 7 July 2006

See also