Difference between revisions of "Cis"

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<math>e^{i\theta} = \cos \theta + i \sin \theta.</math>
 
<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  
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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  
  
 
<math>\mathrm{cis}(r\theta) = (\mathrm{cis}(\theta))^r.</math>
 
<math>\mathrm{cis}(r\theta) = (\mathrm{cis}(\theta))^r.</math>
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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