de Moivre's formula

In mathematics, de Moivre's formula (a.k.a. De Moivre's theorem), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and integer n it holds that

$\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right).\,$

The formula is important because it connects complex numbers (i stands for the imaginary unit (i=√-1)) and trigonometry. The expression cos x + i sin x is sometimes abbreviated to cis x.

By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, 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 a generalization of this formula to find explicit expressions for the nth roots of unity, that is, complex numbers z such that zⁿ = 1.

Derivation

Although historically proven earlier, de Moivre's formula can easily be derived from Euler's formula

$e^{ix} = \cos x + i\sin x\,$

and the exponential law for integer powers

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

Then, by Euler's formula,

$e^{i(nx)} = \cos (nx) + i\sin (nx).\,$

Failure for non-integer powers

De Moivre's formula does not in general hold for non-integer powers. Non-integer powers of a complex number can have many different values, see failure of power and logarithm identities. However there is a generalization that the right hand side expression is one possible value of the power.

The derivation of de Moivre's formula above involves a complex number to the power n. When the power is not an integer, the result is multiple-valued, for example, when n = ½ then:

For x = 0 the formula gives 1^½ = 1

For x = 2π the formula gives 1^½ = −1.

Since the angles 0 and 2π are the same this would give two different values for the same expression. The values 1 and −1 are however both square roots of 1 as the generalization asserts.

No such problem occurs with Euler's formula since there is no identification of different values of its exponent. Euler's formula involves a complex power of a positive real number and this always has a defined value. The corresponding expressions are:

eⁱ⁰ = 1

e^iπ = − 1.

Proof by induction (for integer n)

The truth of de Moivre's theorem can be established by mathematical induction for natural numbers, and extended to all integers from there. Consider S(n):

$(\cos x + i \sin x)^n = \cos (nx) + i \sin (nx), n \in \mathbb{Z}.$

For n > 0, we proceed by mathematical induction. S(1) is clearly true. For our hypothesis, we assume S(k) is true for some natural k. That is, we assume

$\left(\cos x + i \sin x\right)^k = \cos\left(kx\right) + i \sin\left(kx\right). \,$

Now, considering S(k+1):

$\begin{alignat}{2} \left(\cos x+i\sin x\right)^{k+1} & = \left(\cos x+i\sin x\right)^{k} \left(\cos x+i\sin x\right)\\ & = \left[\cos\left(kx\right) + i\sin\left(kx\right)\right] \left(\cos x+i\sin x\right) &&\qquad \text{by the induction hypothesis}\\ & = \cos \left(kx\right) \cos x - \sin \left(kx\right) \sin x + i \left[\cos \left(kx\right) \sin x + \sin \left(kx\right) \cos x\right]\\ & = \cos \left[ \left(k+1\right) x \right] + i\sin \left[ \left(k+1\right) x \right] &&\qquad \text{by the trigonometric identities} \end{alignat}$

We deduce that S(k) implies S(k+1). By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos (0x) + i sin(0x) = 1 +i 0 = 1. Finally, for the negative integer cases, we consider an exponent of -n for natural n.

$\begin{align} \left(\cos x + i\sin x\right)^{-n} & = \left[ \left(\cos x + i\sin x\right)^n \right]^{-1} \\ & = \left[\cos (nx) + i\sin (nx)\right]^{-1} \\ & = \cos(-nx) + i\sin (-nx). \qquad (*) \\ \end{align}$

The equation (*) is a result of the identity $z^{-1} = \frac{\bar{z}}{|z|^2}$, for z = cos nx + i sin nx. Hence, S(n) holds for all integers n.

Formulas for cosine and sine individually

See also: List of trigonometric identities

Being an equality of complex numbers, one necessarily has equality both of the real parts and of the imaginary parts of both members of the equation. If x, and therefore also cos x and sin x, are real numbers, then the identity of these parts can be written using binomial coefficients. This formula was given by 16th century French mathematician Franciscus Vieta:

$\sin nx = \sum_{k=0}^n \binom{n}{k} \cos^kx\,\sin^{n-k}x\,\sin\left(\frac{1}{2}(n-k)\pi\right)$

$\cos nx = \sum_{k=0}^n \binom{n}{k} \cos^kx\,\sin^{n-k}x\,\cos\left(\frac{1}{2}(n-k)\pi\right).$

In each of these two equations, the final trigonometric function equals one or minus one or zero, thus removing half the entries in each of the sums. These equations are in fact even valid for complex values of x, because both sides are entire (that is, holomorphic on the whole complex plane) functions of x, and two such functions that coincide on the real axis necessarily coincide everywhere. Here are the concrete instances of these equations for n = 2 and n = 3:

$\begin{alignat}2 \cos(2x) &= (\cos{x})^2 +((\cos{x})^2-1) &&= 2(\cos{x})^2-1\\ \sin(2x) &= 2(\sin{x})(\cos{x})\\ \cos(3x) &= (\cos{x})^3 +3\cos{x}((\cos{x})^2-1) &&= 4(\cos{x})^3-3\cos{x}\\ \sin(3x) &= 3(\cos{x})^2(\sin{x})-(\sin{x})^3 &&= 3\sin{x}-4(\sin{x})^3.\\ \end{alignat}$

The right hand side of the formula for cos(nx) is in fact the value T_n(cos x) of the Chebyshev polynomial T_n at cos x.

Generalization

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

$\left(\cos z + i\sin z\right)^w$

is a multi-valued function while

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

is not. Therefore one can state that

$\cos (wz) + i \sin (wz) \text{ is one value of } \left(\cos z + i\sin z\right)^w.\,$

Applications

This formula can be used to find the n^th roots of a complex number. This application does not strictly use de Moivre's formula as the power isn't an integer. However considering the right hand side to the power of n will in each case give the same value left hand side.

If z is a complex number, written in polar form as

$z=r\left(\cos x+i\sin x\right),\,$

then

$z^{1/n} = \left[ r\left( \cos x+i\sin x \right) \right]^{1/n} = r^{1/n} \left[ \cos \left( \frac{x+2k\pi}{n} \right) + i\sin \left( \frac{x+2k\pi}{n} \right) \right]$

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.

References

• Abramowitz, Milton & Stegun, Irene A. (1964), Handbook of Mathematical Functions, New York: Dover Publications, p. 74, ISBN 0486612724 .

External links

• De Moivre's Theorem for Trig Identities by Michael Croucher, Wolfram Demonstrations Project.