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 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
.