Difference between revisions of "De Moivre's Theorem"
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&=\frac{1}{(\operatorname{cis} x)^{m}} && \ | &=\frac{1}{(\operatorname{cis} x)^{m}} && \ | ||
&=\frac{1}{\operatorname{cis}(m x)} && \ | &=\frac{1}{\operatorname{cis}(m x)} && \ | ||
− | &=\cos (m x)-i \sin (m x) && \text { rationalization of the denominator } \ | + | &=\cos (m x)-i \sin (m x) &&\text { rationalization of the denominator } \ |
&=\operatorname{cis}(-m x) && \ | &=\operatorname{cis}(-m x) && \ | ||
&=\operatorname{cis}(n x) && | &=\operatorname{cis}(n x) && |
Revision as of 02:08, 6 February 2022
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 .