Difference between revisions of "De Moivre's Theorem"
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<cmath>\begin{align*} | <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 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 | + | & =[\cos (k x)+i \sin (k x)](\cos x+i \sin x) & \text { by our assumption earlier } \ |
& =\cos (k x) \cos x-\sin (k x) \sin x+i[\cos (k x) \sin x+\sin (k x) \cos x] & \ | & =\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 { | + | & =\operatorname{cis}((k+1)(x)) & \text { by various trigonometric identities } |
\end{align*}</cmath> | \end{align*}</cmath> | ||
Revision as of 09:25, 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 .