Difference between revisions of "De Moivre's Theorem"
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(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]] | + | '''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]]: | + | 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> | ||
| − | [[ | + | [[Category:Mathematics]] |
Latest revision as of 23:56, July 14, 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:
