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 [[complex numbers]]. It allows complex numbers in [[polar | + | '''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 <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 == | ||
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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>. Now, the case of <math>n=k+1</math>: | ||
| − | + | <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 the Assumption in Step II } \\ | ||
| + | & =\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 { Various Trigonometric Identities } | ||
| + | \end{align*}</cmath> | ||
:Therefore, the result is true for all positive integers <math>n</math>. | :Therefore, the result is true for all positive integers <math>n</math>. | ||
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*If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer. | *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>\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, [[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>. | |
| + | ==Generalization== | ||
| + | |||
| + | ==See Also== | ||
[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Complex numbers]] | [[Category:Complex numbers]] | ||
Latest revision as of 01:42, 11 January 2024
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.
, for 
, the case is obviously true.- Assume true for the case
. Now, the case of 
:
. Now, the case of 
: & \text { by the Assumption in Step II } \\ & =\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 { Various Trigonometric Identities } \end{align*}](http://latex.artofproblemsolving.com/d/e/6/de60109242847b17882883d070585f1a3d59ab96.png)
- Therefore, the result is true for all positive integers
.
- If
, the formula holds true because 
. Since 
, the equation holds true.
, the formula holds true because 
. Since 
, the equation holds true.- If
, one must consider 
when 
is a positive integer.
, 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 identity states that 
. This extends De Moivre's theorem to all 
.
