Difference between revisions of "De Moivre's Theorem"

Revision as of 12:13, 10 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 $x\in\mathbb{R}$ and $n\in\mathbb{N}$ , $\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)$ .

Proof

This is one proof of De Moivre's theorem by induction.

If $n>0$ , for $n=1$ , the case is obviously true.

Assume true for the case $n=k$ . Now, the case of $n=k+1$ :

Therefore, the result is true for all positive integers $n$ .

If $n=0$ , the formula holds true because $\cos(0x)+i\sin (0x)=1+i0=1$ . Since $z^0=1$ , the equation holds true.

If $n<0$ , one must consider $n=-m$ when $m$ is a positive integer.

And thus, the formula proves true for all integral values of $n$ . $\Box$

Note that from the functional equation $f(x) = f(nx)$ where $f(x) = \cos x + i\sin x$ , we see that $f(x)$ behaves like an exponential function. Indeed, by Euler's formula ($e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. Unknown error_msg)). This extends De Moivre's theorem to all $n\in \mathbb{R}$ .

@@ 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 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 form]] to be easily raised to certain powers. It states that for <math>x\in\mathbb{R}</math> and <math>n\in\mathbb{N}</math>, <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>.
 == Proof ==
@@ Line 22: / Line 22: @@
 And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math>
-By [[Euler's formula]] (<math>e^{ix} = \cos x+i\sin x\right</math>), this can be extended to all real numbers <math>n</math>.
+Note that from the functional equation <math>f(x) = 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, by [[Euler's formula]] (<math>e^{ix} = \cos x+i\sin x\right</math>). This extends De Moivre's theorem to all <math>n\in \mathbb{R}</math>.
 [[Category:Theorems]]
 [[Category:Complex numbers]]

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Difference between revisions of "De Moivre's Theorem"

Revision as of 12:13, 10 September 2008

Proof