NationMaster - Encyclopedia: De Moivre's formula

de Moivre's formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that Abraham de Moivre. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The integers are commonly denoted by the above symbol. ...

$left(cos x+isin xright)^n=cosleft(nxright)+isinleft(nxright).,$

The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. The expression "cos x + i sin x" is sometimes abbreviated to "cis x". The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ... All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...

By expanding the left hand side and then comparing the real and imaginary parts, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x). Furthermore, one can use this formula to find explicit expressions for the n-th roots of unity, that is, complex numbers z such that zⁿ = 1. In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. ...

1 Derivation
2 Proof by induction
3 Generalization
4 Applications
5 See also

Derivation

Although historically proved earlier, de Moivre's formula can easily be derived from Euler's formula This article is about the Eulers formula in complex analysis. ...

$e^{ix} = cos x + isin x,$

and the exponential law The exponential function is one of the most important functions in mathematics. ...

$left( e^{ix} right)^n = e^{inx} .,$

Proof by induction

We consider three cases.

For n > 0, we proceed by mathematical induction. When n = 1, the result is clearly true. For our hypothesis, we assume the result is true for some positive integer k. That is, we assume Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

$left(cos x + i sin xright)^k = cosleft(kxright) + i sinleft(kxright). ,$

Now, considering the case n = k + 1:

$begin{alignat}{2} left(cos x+isin xright)^{k+1} & = left(cos x+isin xright)^{k} left(cos x+isin xright) & = left[cosleft(kxright) + isinleft(kxright)right] left(cos x+isin xright) qquad mbox{by the induction hypothesis} & = cos left(kxright) cos x - sin left(kxright) sin x + i left[cos left(kxright) sin x + sin left(kxright) cos xright] & = cos left[ left(k+1right) x right] + isin left[ left(k+1right) x right] qquad mbox{by the trigonometric identities} end{alignat}$

We deduce that the result is true for n = k + 1 when it is true for n = k. By the Principle of Mathematical Induction it follows that the result is true for all positive integers n.

When n = 0 the formula is true since $cos(0 x) + i sin(0 x) = 1 + i 0 = 1$ , and (by convention) $z 0 = 1$ .

When n < 0, we consider a positive integer m such that n = −m. So

$begin{alignat}{2} left(cos x + isin xright)^{n} & = left(cos x + isin xright)^{-m} & = frac{1}{left(cos x + isin xright)^{m}} & = frac{1}{left(cos mx + isin mxright)} & = cosleft(mxright) - isinleft(mxright) & = cosleft(-mxright) + isinleft(-mxright) & = cosleft(nxright) + isinleft(nxright). end{alignat}$

Hence, the theorem is true for all integer values of n. Q.E.D. Look up QED in Wiktionary, the free dictionary. ...

Generalization

The formula is actually true in a more general setting than stated above: if z and w are complex numbers, then

$left(cos z + isin zright)^w$

is a multivalued function while This diagram does not represent a true function, because the element 3 in X is associated with two elements, b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...

$cos (wz) + i sin (wz),$

is not. Therefore one can state that

$cos (wz) + i sin (wz) ,$ is one value of $left(cos z + isin zright)^w,$ .

A plot on the complex plane of the cubed roots of 1.

Image File history File links Size of this preview: 229 Ã— 213 pixelsFull resolution (229 Ã— 213 pixel, file size: 9 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): De Moivres formula ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

Applications

This formula can be used to find the $n t h$ roots of a complex number. If $z$ is a complex number, written in polar form as

$z=rleft(cos x+isin xright),,$

then

$begin{align} z^{1/n} & = left[ rleft( cos x+isin x right) right]^{1/n} & = r^{1/n} left[ cos left( frac{x+2kpi}{n} right) + isin left( frac{x+2kpi}{n} right) right] end{align}$

where $k$ is an integer, to get the $n$ different roots of $z$ one only needs to consider values of $k$ from $0$ to $n - 1$ .

Contents

Proof by induction

Generalization

Applications

See also