Difference between revisions of "De Moivre's Theorem"

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(New page: De Moivre’s Theorem is a fundamental statement of complex analysis, where ''i'' represents the square root of (-1): :<math>\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\le...)
 
(Don't need to define "i"--it's fundamental. It's primordial. The fact that it's one of the square roots of -1 (and -i is the other) is a theorem.)
 
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De Moivre’s Theorem is a fundamental statement of [[complex analysis]], where ''i'' represents the square root of (-1):
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'''De Moivre’s Theorem''' is a fundamental statement of [[complex analysis]]:
  
 
:<math>\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right)\,</math>
 
:<math>\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right)\,</math>
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:<math>\left( e^{ix} \right)^n = e^{inx} \,</math>
 
:<math>\left( e^{ix} \right)^n = e^{inx} \,</math>
  
Therefore from [[Euler's formula]]:
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Therefore, from [[Euler's formula]]:
  
 
:<math>e^{i(nx)} = \cos(nx) + i\sin(nx)\,</math>
 
:<math>e^{i(nx)} = \cos(nx) + i\sin(nx)\,</math>
 
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[[Category:Mathematics]]
[[category:mathematics]]
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Latest revision as of 18:56, 14 July 2018

De Moivre’s Theorem is a fundamental statement of complex analysis:

Extension of Euler's formula

De Moivre's formula is a trivial extension of Euler's formula:

Because

Therefore, from Euler's formula: