Difference between revisions of "De Moivre's Theorem"
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& =[\cos (k x)+i \sin (k x)](\cos x+i \sin x) & \text { by the Assumption in Step II } \ | & =[\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] & \ | & =\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) & \text { Various Trigonometric Identities } | + | & =\operatorname{cis}((k+1)(x)) & \text { Various Trigonometric Identities } |
\end{align*}</cmath> | \end{align*}</cmath> | ||
Revision as of 11:29, 2 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.
- 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 identity states that . This extends De Moivre's theorem to all .