Difference between revisions of "De Moivre's Theorem"
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And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math> | And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math> | ||
| − | Note that from the functional equation <math>f(x)^n = f(nx)</math> where <math>f(x) = \cos x + i\sin x</math>, we see that <math>f(x)</math> behaves like an exponential function. Indeed, [[Euler's | + | Note that from the functional equation <math>f(x)^n = f(nx)</math> where <math>f(x) = \cos x + i\sin x</math>, we see that <math>f(x)</math> behaves like an exponential function. Indeed, [[Euler's identity]] states that <math>e^{ix} = \cos x+i\sin x</math>. This extends De Moivre's theorem to all <math>n\in \mathbb{R}</math>. |
==Generalization== | ==Generalization== | ||
Revision as of 04:04, 29 April 2018
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.
, for 
, the case is obviously true.- Assume true for the case
. Now, the case of 
:
. Now, the case of 
:
- Therefore, the result is true for all positive integers
.
- If
, the formula holds true because 
. Since 
, the equation holds true.
, the formula holds true because 
. Since 
, the equation holds true.- If
, one must consider 
when 
is a positive integer.
, 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 
.
