Difference between revisions of "De Moivre's Theorem"
(Tag: Undo) |
|||
(7 intermediate revisions by 3 users not shown) | |||
Line 11: | Line 11: | ||
(\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 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)+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> | ||
Line 22: | Line 22: | ||
<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) | + | &=\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> | ||
Line 36: | Line 36: | ||
==Generalization== | ==Generalization== | ||
− | + | ==See Also== | |
[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Complex numbers]] | [[Category:Complex numbers]] |
Revision as of 01:42, 11 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
.