Difference between revisions of "De Moivre's Theorem"
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<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | (\operatorname{cis} x)^{n} &=(\operatorname{cis} x)^{-m} & \\ | + | (\operatorname{cis} x)^{n} &=(\operatorname{cis} x)^{-m} && \\ |
− | &=\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) && |
\end{align*}</cmath> | \end{align*}</cmath> | ||
Revision as of 03:08, 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
.