De Moivre's formula

Template:Use American English Template:Short description Template:Not to be confused

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number Template:Mvar and integer Template:Mvar it is the case that <math display="block">\big(\cos x + i \sin x\big)^n = \cos nx + i \sin nx,</math> where Template:Mvar is the imaginary unit (Template:Math). The formula is named after Abraham de Moivre,<ref>Template:Cite journal

English translation by Richard J. Pulskamp (2009)

On p. 2370 de Moivre stated that if a series has the form <math>ny + \tfrac{1 - nn}{2 \times 3}ny^3 + \tfrac{1 - nn}{2 \times 3} \tfrac{9 - nn}{4 \times 5}ny^5 + \tfrac{1 - nn}{2 \times 3} \tfrac{9 - nn}{4 \times 5} \tfrac{25 - nn}{6 \times 7}ny^7 + \cdots = a</math> , where n is any given odd integer (positive or negative) and where y and a can be functions, then upon solving for y, the result is equation (2) on the same page: <math> y = \tfrac{1}{2}\sqrt[n]{a + \sqrt{aa-1}} + \tfrac{1}{2}\sqrt[n]{a - \sqrt{aa-1}}</math>. If y = cos x and a = cos nx , then the result is <math> \cos x = \tfrac{1}{2} (\cos(nx) + i\sin(nx))^{1/n} + \tfrac{1}{2}(\cos(nx) - i\sin(nx))^{1/n}</math>

In 1676, Isaac Newton found the relation between two chords that were in the ratio of n to 1; the relation was expressed by the series above. The series appears in a letter — Epistola prior D. Issaci Newton, Mathescos Professoris in Celeberrima Academia Cantabrigiensi; … — of 13 June 1676 from Isaac Newton to Henry Oldenburg, secretary of the Royal Society; a copy of the letter was sent to Gottfried Wilhelm Leibniz. See p. 106 of: Template:Cite book
In 1698, de Moivre derived the same series. See: Template:Cite journal; see p 192.
In 1730, de Moivre explicitly considered the case where the functions are cos θ and cos nθ. See: Template:Cite book From p. 1: "Lemma 1. Si sint l & x cosinus arcuum duorum A & B, quorum uterque eodem radio 1 describatur, quorumque prior sit posterioris multiplex in ea ratione quam habet numerus n ad unitatem, tunc erit <math>x = \tfrac{1}{2}\sqrt[n]{l + \sqrt{ll-1}} + \tfrac{1}{2}\tfrac{1}{\sqrt[n]{l + \sqrt{ll-1}}}</math>." (If l and x are cosines of two arcs A and B both of which are described by the same radius 1 and of which the former is a multiple of the latter in that ratio as the number n has to 1, then it will be [true that] <math>x = \tfrac{1}{2}\sqrt[n]{l + \sqrt{ll-1}} + \tfrac{1}{2}\tfrac{1}{\sqrt[n]{l + \sqrt{ll-1}}}</math>.) So if arc A = n × arc B, then l = cos A = cos nB and x = cos B. Hence <math> \cos B = \tfrac{1}{2} (\cos(nB) + \sqrt{-1}\sin(nB))^{1/n} + \tfrac{1}{2}(\cos(nB) + \sqrt{-1}\sin(nB))^{-1/n} </math>

Example

For <math> x = 30^\circ</math> and <math> n = 2</math>, de Moivre's formula asserts that <math display=block>\left(\cos(30^\circ) + i \sin(30^\circ)\right)^2 = \cos(2 \cdot 30^\circ) + i \sin (2 \cdot 30^\circ),</math> or equivalently that <math display=block>\left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^2 = \frac{1}{2} + \frac{i\sqrt{3}}{2}.</math> In this example, it is easy to check the validity of the equation by multiplying out the left side.

Relation to Euler's formula

De Moivre's formula is a precursor to Euler's formula <math display=block>e^{ix} = \cos x + i\sin x,</math> with Template:Mvar expressed in radians rather than degrees, which establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers

<math>\left( e^{ix} \right)^n = e^{inx}, </math>

since Euler's formula implies that the left side is equal to <math>\left(\cos x + i\sin x\right)^n</math> while the right side is equal to <math>\cos nx + i\sin nx.</math>

Proof by induction

The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer Template:Mvar, call the following statement Template:Math:

For Template:Math, we proceed by mathematical induction. Template:Math is clearly true. For our hypothesis, we assume Template:Math is true for some natural Template:Mvar. That is, we assume

<math>\left(\cos x + i \sin x\right)^k = \cos kx + i \sin kx. </math>

Now, considering Template:Math:

<math>\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 kx + i\sin kx \right) \left(\cos x+i\sin x\right) &&\qquad \text{by the induction hypothesis}\\
& = \cos kx \cos x - \sin kx \sin x + i \left(\cos kx \sin x + \sin kx \cos x\right)\\
& = \cos ((k+1)x) + i\sin ((k+1)x) &&\qquad \text{by the trigonometric identities}

\end{alignat}</math>

See angle sum and difference identities.

We deduce that Template:Math implies Template:Math. By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, Template:Math is clearly true since Template:Math. Finally, for the negative integer cases, we consider an exponent of Template:Math for natural Template:Mvar.

<math>\begin{align}

\left(\cos x + i\sin x\right)^{-n} & = \big( \left(\cos x + i\sin x\right)^n \big)^{-1} \\
& = \left(\cos nx + i\sin nx\right)^{-1} \\
& =  \cos nx - i\sin nx \qquad\qquad(*)\\
& = \cos(-nx) + i\sin (-nx).\\

\end{align}</math> The equation (*) is a result of the identity

for Template:Math. Hence, Template:Math holds for all integers Template:Mvar.

Formulae for cosine and sine individually

Template:See also For 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 Template:Mvar, and therefore also Template:Math and Template:Math, are real numbers, then the identity of these parts can be written using binomial coefficients. This formula was given by 16th century French mathematician François Viète:

<math>\begin{align}

\sin nx &= \sum_{k=0}^n \binom{n}{k} (\cos x)^k\,(\sin x)^{n-k}\,\sin\frac{(n-k)\pi}{2} \\ \cos nx &= \sum_{k=0}^n \binom{n}{k} (\cos x)^k\,(\sin x)^{n-k}\,\cos\frac{(n-k)\pi}{2}. \end{align}</math>

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 valid even for complex values of Template:Mvar, because both sides are entire (that is, holomorphic on the whole complex plane) functions of Template:Mvar, and two such functions that coincide on the real axis necessarily coincide everywhere. Here are the concrete instances of these equations for Template:Math and Template:Math:

<math>\begin{alignat}{2}

\cos 2x &= \left(\cos x\right)^2 +\left(\left(\cos x\right)^2-1\right)        &{}={}& 2\left(\cos x\right)^2-1       \\
\sin 2x &= 2\left(\sin x\right)\left(\cos x\right)                            &     &                                \\
\cos 3x &= \left(\cos x\right)^3 +3\cos x\left(\left(\cos x\right)^2-1\right) &{}={}& 4\left(\cos x\right)^3-3\cos x \\
\sin 3x &= 3\left(\cos x\right)^2\left(\sin x\right)-\left(\sin x\right)^3    &{}={}& 3\sin x-4\left(\sin x\right)^3.

\end{alignat}</math>

The right-hand side of the formula for Template:Math is in fact the value Template:Math of the Chebyshev polynomial Template:Math at Template:Math.

Failure for non-integer powers, and generalization

De Moivre's formula does not hold for non-integer powers. The derivation of de Moivre's formula above involves a complex number raised to the integer power Template:Mvar. If a complex number is raised to a non-integer power, the result is multiple-valued (see failure of power and logarithm identities).

Roots of complex numbers

A modest extension of the version of de Moivre's formula given in this article can be used to find [[nth root|the Template:Mvar-th roots]] of a complex number for a non-zero integer Template:Mvar. If Template:Mvar is a complex number, written in polar form as <math display="block"> z=r\left(\cos x+i\sin x\right), </math> then the Template:Mvar-th roots of Template:Mvar are given by <math display="block"> r^\frac1n \left( \cos \frac{x+2\pi k}{n} + i\sin \frac{x+2\pi k}{n} \right) </math> where Template:Mvar varies over the integer values from 0 to Template:Math. This formula is also sometimes known as de Moivre's formula.<ref>Template:Springer</ref>

Complex numbers raised to an arbitrary power

Generally, if <math>z=r\left(\cos x+i\sin x\right)</math> (in polar form) and Template:Mvar are arbitrary complex numbers, then the set of possible values is <math display=block>z^w = r^w \left(\cos x + i\sin x\right)^w = \lbrace r^w \cos(xw + 2\pi kw) + i r^w \sin(xw + 2\pi kw) | k \in \mathbb{Z}\rbrace\,.</math> (Note that if Template:Mvar is a rational number that equals Template:Math in lowest terms then this set will have exactly Template:Mvar distinct values rather than infinitely many. In particular, if Template:Mvar is an integer then the set will have exactly one value, as previously discussed.) In contrast, de Moivre's formula gives <math display=block>r^w (\cos xw + i\sin xw)\,,</math> which is just the single value from this set corresponding to Template:Math.

Analogues in other settings

Hyperbolic trigonometry

Since Template:Math, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all integers Template:Mvar,

<math>(\cosh x + \sinh x)^n = \cosh nx + \sinh nx.</math>

If Template:Mvar is a rational number (but not necessarily an integer), then Template:Math will be one of the values of Template:Math.<ref>Template:Cite journal</ref>

Extension to complex numbers

For any integer Template:Mvar, the formula holds for any complex number <math>z=x+iy</math>

where

<math>\begin{align} \cos z = \cos(x + iy) &= \cos x \cosh y - i \sin x \sinh y\, , \\

\sin z = \sin(x + iy) &= \sin x \cosh y + i \cos x \sinh y\, . \end{align}</math>

Quaternions

To find the roots of a quaternion there is an analogous form of de Moivre's formula. A quaternion in the form

<math>q = d + a\mathbf{\hat i} + b\mathbf{\hat j} + c\mathbf{\hat k}</math>

can be represented in the form

<math>q = k(\cos \theta + \varepsilon \sin \theta) \qquad \mbox{for } 0 \leq \theta < 2 \pi.</math>

In this representation,

and the trigonometric functions are defined as

<math>\cos \theta = \frac{d}{k} \quad \mbox{and} \quad \sin \theta = \pm \frac{\sqrt{a^2 + b^2 + c^2}}{k}.</math>

In the case that Template:Math,

<math>\varepsilon = \pm \frac{a\mathbf{\hat i} + b\mathbf{\hat j} + c\mathbf{\hat k}}{\sqrt{a^2 + b^2 + c^2}},</math>

that is, the unit vector. This leads to the variation of De Moivre's formula:

<math>q^n = k^n(\cos n \theta + \varepsilon \sin n \theta).</math><ref>Template:Cite journal</ref>

Example

To find the cube roots of

<math>Q = 1 + \mathbf{\hat i} + \mathbf{\hat j}+ \mathbf{\hat k},</math>

write the quaternion in the form

<math>Q = 2\left(\cos \frac{\pi}{3} + \varepsilon \sin \frac{\pi}{3}\right) \qquad \mbox{where } \varepsilon = \frac{\mathbf{\hat i} + \mathbf{\hat j}+ \mathbf{\hat k}}{\sqrt 3}.</math>

Then the cube roots are given by:

<math>\sqrt[3]{Q} = \sqrt[3]{2}(\cos \theta + \varepsilon \sin \theta) \qquad \mbox{for } \theta = \frac{\pi}{9}, \frac{7\pi}{9}, \frac{13\pi}{9}.</math>

2 × 2 matrices

With matrices, <math>\begin{pmatrix}\cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{pmatrix}^n=\begin{pmatrix}\cos n\phi & -\sin n\phi \\ \sin n\phi & \cos n\phi \end{pmatrix}</math> when Template:Mvar is an integer. This is a direct consequence of the isomorphism between the matrices of type <math>\begin{pmatrix}a & -b \\ b & a \end{pmatrix}</math> and the complex plane.

References

Template:Cite book.

Template:Reflist

External links

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

Template:Spoken Wikipedia