Difference between revisions of "De Moivre's Theorem"
m |
|||
Line 1: | Line 1: | ||
+ | {{WotWAnnounce|week=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 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 17:53, 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.
- 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 .
By Euler's formula ($e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. ! Missing delimiter (. inserted).)), this can be extended to all real numbers .