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 + | + | '''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 form|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]] | ||
Latest revision as of 19:59, 11 July 2023
Cis notation is a polar notation for complex numbers. For all complex numbers 
, we can write 
. Notice that 
is made up by the first letter of 
, 
, and the first letter of 
.
Once one gets used to the notation, it is almost always preferred to write 
rather than 
, as Euler's formula states that
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
This is more easily understood in the complex exponential form:
