Difference between revisions of "De Moivre's Theorem"
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'''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It allows complex numbers in [[polar coordinates|polar]] form to be easily raised to certain powers. It states that for an <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math> | '''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It allows complex numbers in [[polar coordinates|polar]] form to be easily raised to certain powers. It states that for an <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math> | ||
Revision as of 16:53, 6 September 2008
| This is an AoPSWiki Word of the Week for September 5- September 11 |
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 an 
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 
. 
By Euler's formula ($e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. Unknown error_msg)), this can be extended to all real numbers .
