Difference between revisions of "De Moivre's Theorem"
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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> | ||
− | 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, | + | 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 formula]] states that <math>e^{ix} = \cos x+i\sin x\right</math>. This extends De Moivre's theorem to all <math>n\in \mathbb{R}</math>. |
[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Complex numbers]] | [[Category:Complex numbers]] |
Revision as of 12:14, 10 September 2008
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DeMoivre'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 case is obviously true.
- Assume true for the case . Now, the case of :
- 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 .
Note that from the functional equation where , we see that behaves like an exponential function. Indeed, Euler's formula states that $e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. Unknown error_msg). This extends De Moivre's theorem to all .