Difference between revisions of "Cis"

 
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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=\mathrm{cis }(\theta)=\cos \theta + i\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>.
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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>.
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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
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<math>e^{i\theta} = \cos \theta + i \sin \theta.</math>
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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
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<math>\mathrm{cis}(r\theta) = (\mathrm{cis}(\theta))^r.</math>
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This is more easily understood in the complex exponential form:
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<math>e^{i(r\theta)} = (e^{i\theta})^r.</math>
  
 
== See also ==
 
== See also ==
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* [[Complex numbers]]
 
* [[Complex numbers]]
 
* [[Polar form]]
 
* [[Polar form]]
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[[Category:Complex numbers]]

Revision as of 00:12, 23 September 2017

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:

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

See also