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 == | |
+ | This is one proof of de Moivre's theorem by [[induction]]. | ||
+ | *If <math>n\ge0</math>: | ||
− | + | :If <math>n=0</math>, the formula holds true because <math>\cos(0x)+i\sin(0x)=1+0i=1=z^0.</math> | |
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− | + | :Assume the formula is true for <math>n=k</math>. Now, consider <math>n=k+1</math>: | |
− | Assume true for | ||
− | === | + | <cmath>\begin{align*} |
− | + | (\cos x+i \sin x)^{k+1} & =(\cos x+i \sin x)^{k}(\cos x+i \sin x) & \text { by Exponential laws } \ | |
+ | & =[\cos (k x)+i \sin (k x)](\cos x+i \sin x) & \text { by our assumption } \ | ||
+ | & =\cos (k x) \cos x-\sin (k x) \sin x+i[\cos (k x) \sin x+\sin (k x) \cos x] & \ | ||
+ | & =\operatorname{cis}((k+1)(x)) & \text { by various trigonometric identities } | ||
+ | \end{align*}</cmath> | ||
− | + | :Therefore, the result is true for all nonnegative integers <math>n</math>. | |
− | + | *If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer. | |
− | + | <cmath>\begin{align*} | |
− | + | (\operatorname{cis} x)^{n} &=(\operatorname{cis} x)^{-m} \ | |
+ | &=\frac{1}{(\operatorname{cis} x)^{m}} \ | ||
+ | &=\frac{1}{\operatorname{cis}(m x)} \ | ||
+ | &=\cos (m x)-i \sin (m x) & \text { rationalization of the denominator } \ | ||
+ | &=\operatorname{cis}(-m x) \ | ||
+ | &=\operatorname{cis}(n x) | ||
+ | \end{align*}</cmath> | ||
− | + | And thus, the formula proves true for all integral values of <math>n</math>. <math>\blacksquare</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>. |
− | + | ==See Also== | |
+ | [[Category:Theorems]] | ||
+ | [[Category:Complex numbers]] |
Latest revision as of 09:49, 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 :
- If , the formula holds true because
- Assume the formula is true for . Now, consider :
- Therefore, the result is true for all nonnegative integers .
- 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 .