Difference between revisions of "De Moivre's Theorem"
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− | ''' | + | '''De Moivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It allows complex numbers in [[polar form]] to be easily raised to certain powers. It states that for <math>x\in\mathbb{R}</math> and <math>n\in\mathbb{Z}</math>, <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>. |
== Proof == | == Proof == | ||
This is one proof of de Moivre's theorem by [[induction]]. | This is one proof of de Moivre's theorem by [[induction]]. | ||
− | *If <math>n>0</math> | + | *If <math>n>0</math>: |
− | :Assume true for the case <math>n=k</math>. Now, | + | :For <math>n=1</math>, the proposition is obviously true. |
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+ | :Assume true for the case <math>n=k</math>. Now, for <math>n=k+1</math>: | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
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And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math> | And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math> | ||
+ | |||
+ | ==Generalization== | ||
Note that from the functional equation <math>f(x)^n = f(nx)</math> where <math>f(x) = \cos x + i\sin x</math>, we see that <math>f(x)</math> behaves like an exponential function. Indeed, [[Euler's identity]] states that <math>e^{ix} = \cos x+i\sin x</math>. This extends de Moivre's theorem to all <math>n\in \mathbb{R}</math>. | Note that from the functional equation <math>f(x)^n = f(nx)</math> where <math>f(x) = \cos x + i\sin x</math>, we see that <math>f(x)</math> behaves like an exponential function. Indeed, [[Euler's identity]] states that <math>e^{ix} = \cos x+i\sin x</math>. This extends de Moivre's theorem to all <math>n\in \mathbb{R}</math>. | ||
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==See Also== | ==See Also== | ||
[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Complex numbers]] | [[Category:Complex numbers]] |
Revision as of 09:10, 31 August 2024
De Moivre's Theorem is a very useful theorem in the mathematical fields of complex numbers. It allows complex numbers in polar form to be easily raised to certain powers. It states that for and , .
Proof
This is one proof of de Moivre's theorem by induction.
- If :
- For , the proposition is obviously true.
- Assume true for the case . Now, for :
- Therefore, the result is true for all positive integers .
- If , the formula holds true because . Since , the equation holds true.
- If , one must consider when is a positive integer.
And thus, the formula proves true for all integral values of .
Generalization
Note that from the functional equation where , we see that behaves like an exponential function. Indeed, Euler's identity states that . This extends de Moivre's theorem to all .