Difference between revisions of "De Moivre's Theorem"
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− | DeMoivre's Theorem is a very useful theorem in the mathematical fields of [[ | + | '''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It allows complex numbers in [[polar coordinates|polar]] form to be easily raised to certain powers. It states that for an <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math> |
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− | <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math> | ||
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== Proof == | == Proof == | ||
− | This is one proof of | + | This is one proof of De Moivre's theorem by [[induction]]. |
− | + | *If <math>n>0</math>, for <math>n=1</math>, the case is obviously true. | |
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− | + | :Assume true for the case <math>n=k</math>. Now, the case of <math>n=k+1</math>: | |
− | Assume true for the case <math>n=k</math>. | ||
− | + | :[[Image:DeMoivreInductionP1.gif]] | |
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− | + | :Therefore, the result is true for all positive integers <math>n</math>. | |
− | + | *If <math>n=0</math>, the formula holds true because <math>\cos(0x)+i\sin (0x)=1+i0=1</math>. Since <math>z^0=1</math>, the equation holds true. | |
− | + | *If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer. | |
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− | + | :[[Image:DeMoivreInductionP2.gif]] | |
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− | + | And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math> | |
− | [[ | + | By [[Euler's formula]] (<math>e^{ix} = \cos x+i\sin x\right</math>), this can be extended to all real numbers <math>n</math>. |
− | + | [[Category:Theorems]] |
Revision as of 20:10, 23 November 2007
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 an
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 .
By Euler's formula ($e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. Unknown error_msg)), this can be extended to all real numbers .