De Moivre's formula

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number $x$ and integer $n$ it is the case that ${\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx,$ where $i$ is the imaginary unit ( $i 2 = -1$ ). The formula is named after Abraham de Moivre,^[1] although he never stated it in his works.^[2] The expression $cos x + i sin x$ is sometimes abbreviated to $cis x$ .

The formula is important because it connects complex numbers and trigonometry. 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 $sin x$ .

As written, the formula is not valid for non-integer powers $n$ . However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the $n$ th roots of unity, that is, complex numbers $z$ such that $z n = 1$ .

Using the standard extensions of the sine and cosine functions to complex numbers, the formula is valid even when $x$ is an arbitrary complex number.

^
Moivre, Ab. de (1707). "Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, & superiorum, ad infinitum usque pergendo, in termimis finitis, ad instar regularum pro cubicis quae vocantur Cardani, resolutio analytica" [Of certain equations of the third, fifth, seventh, ninth, & higher power, all the way to infinity, by proceeding, in finite terms, in the form of rules for cubics which are called by Cardano, resolution by analysis.]. Philosophical Transactions of the Royal Society of London (in Latin). 25 (309): 2368–2371. doi:10.1098/rstl.1706.0037. S2CID 186209627.
- English translation by Richard J. Pulskamp (2009)
On p. 2370 de Moivre stated that if a series has the form $ny+{\tfrac {1-nn}{2\times 3}}ny^{3}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}ny^{5}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}{\tfrac {25-nn}{6\times 7}}ny^{7}+\cdots =a$ $ny+{\tfrac {1-nn}{2\times 3}}ny^{3}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}ny^{5}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}{\tfrac {25-nn}{6\times 7}}ny^{7}+\cdots =a$ , 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: $y={\tfrac {1}{2}}{\sqrt[{n}]{a+{\sqrt {aa-1}}}}+{\tfrac {1}{2}}{\sqrt[{n}]{a-{\sqrt {aa-1}}}}$ $y={\tfrac {1}{2}}{\sqrt[{n}]{a+{\sqrt {aa-1}}}}+{\tfrac {1}{2}}{\sqrt[{n}]{a-{\sqrt {aa-1}}}}$ . If y = cos x and a = cos nx , then the result is $\cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}$ $\cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}$
- 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: Biot, J.-B.; Lefort, F., eds. (1856). Commercium epistolicum J. Collins et aliorum de analysi promota, etc: ou … (in Latin). Paris, France: Mallet-Bachelier. pp. 102–112.
- In 1698, de Moivre derived the same series. See: de Moivre, A. (1698). "A method of extracting roots of an infinite equation". Philosophical Transactions of the Royal Society of London. 20 (240): 190–193. doi:10.1098/rstl.1698.0034. S2CID 186214144.; see p 192.
- In 1730, de Moivre explicitly considered the case where the functions are cos θ and cos nθ. See: Moivre, A. de (1730). Miscellanea Analytica de Seriebus et Quadraturis (in Latin). London, England: J. Tonson & J. Watts. p. 1. 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 $x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}$ ." (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] $x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}$ .) So if arc A = n × arc B, then l = cos A = cos nB and x = cos B. Hence $\cos B={\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{1/n}+{\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{-1/n}$
See also:
- Cantor, Moritz (1898). Vorlesungen über Geschichte der Mathematik [Lectures on the History of Mathematics]. Bibliotheca mathematica Teuberiana, Bd. 8-9 (in German). Vol. 3. Leipzig, Germany: B.G. Teubner. p. 624.
- Braunmühl, A. von (1901). "Zur Geschichte der Entstehung des sogenannten Moivreschen Satzes" [On the history of the origin of the so-called Moivre theorem]. Bibliotheca Mathematica. 3rd series (in German). 2: 97–102.; see p. 98.
^ Lial, Margaret L.; Hornsby, John; Schneider, David I.; Callie J., Daniels (2008). College Algebra and Trigonometry (4th ed.). Boston: Pearson/Addison Wesley. p. 792. ISBN 9780321497444.

[1] Moivre, Ab. de (1707). "Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, & superiorum, ad infinitum usque pergendo, in termimis finitis, ad instar regularum pro cubicis quae vocantur Cardani, resolutio analytica" [Of certain equations of the third, fifth, seventh, ninth, & higher power, all the way to infinity, by proceeding, in finite terms, in the form of rules for cubics which are called by Cardano, resolution by analysis.]. Philosophical Transactions of the Royal Society of London (in Latin). 25 (309): 2368–2371. doi:10.1098/rstl.1706.0037. S2CID 186209627.
English translation by Richard J. Pulskamp (2009)
On p. 2370 de Moivre stated that if a series has the form $ny+{\tfrac {1-nn}{2\times 3}}ny^{3}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}ny^{5}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}{\tfrac {25-nn}{6\times 7}}ny^{7}+\cdots =a$ , 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: $y={\tfrac {1}{2}}{\sqrt[{n}]{a+{\sqrt {aa-1}}}}+{\tfrac {1}{2}}{\sqrt[{n}]{a-{\sqrt {aa-1}}}}$ . If y = cos x and a = cos nx , then the result is $\cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}$

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: Biot, J.-B.; Lefort, F., eds. (1856). Commercium epistolicum J. Collins et aliorum de analysi promota, etc: ou … (in Latin). Paris, France: Mallet-Bachelier. pp. 102–112.

In 1698, de Moivre derived the same series. See: de Moivre, A. (1698). "A method of extracting roots of an infinite equation". Philosophical Transactions of the Royal Society of London. 20 (240): 190–193. doi:10.1098/rstl.1698.0034. S2CID 186214144.; see p 192.

In 1730, de Moivre explicitly considered the case where the functions are cos θ and cos nθ. See: Moivre, A. de (1730). Miscellanea Analytica de Seriebus et Quadraturis (in Latin). London, England: J. Tonson & J. Watts. p. 1. 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 $x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}$ ." (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] $x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}$ .) So if arc A = n × arc B, then l = cos A = cos nB and x = cos B. Hence $\cos B={\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{1/n}+{\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{-1/n}$
See also:
Cantor, Moritz (1898). Vorlesungen über Geschichte der Mathematik [Lectures on the History of Mathematics]. Bibliotheca mathematica Teuberiana, Bd. 8-9 (in German). Vol. 3. Leipzig, Germany: B.G. Teubner. p. 624.

Braunmühl, A. von (1901). "Zur Geschichte der Entstehung des sogenannten Moivreschen Satzes" [On the history of the origin of the so-called Moivre theorem]. Bibliotheca Mathematica. 3rd series (in German). 2: 97–102.; see p. 98.

[2] English translation by Richard J. Pulskamp (2009)

[3] 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: Biot, J.-B.; Lefort, F., eds. (1856). Commercium epistolicum J. Collins et aliorum de analysi promota, etc: ou … (in Latin). Paris, France: Mallet-Bachelier. pp. 102–112.

[4] In 1698, de Moivre derived the same series. See: de Moivre, A. (1698). "A method of extracting roots of an infinite equation". Philosophical Transactions of the Royal Society of London. 20 (240): 190–193. doi:10.1098/rstl.1698.0034. S2CID 186214144.; see p 192.

[5] In 1730, de Moivre explicitly considered the case where the functions are cos θ and cos nθ. See: Moivre, A. de (1730). Miscellanea Analytica de Seriebus et Quadraturis (in Latin). London, England: J. Tonson & J. Watts. p. 1. 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 $x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}$ ." (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] $x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}$ .) So if arc A = n × arc B, then l = cos A = cos nB and x = cos B. Hence $\cos B={\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{1/n}+{\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{-1/n}$

[6] Cantor, Moritz (1898). Vorlesungen über Geschichte der Mathematik [Lectures on the History of Mathematics]. Bibliotheca mathematica Teuberiana, Bd. 8-9 (in German). Vol. 3. Leipzig, Germany: B.G. Teubner. p. 624.

[7] Braunmühl, A. von (1901). "Zur Geschichte der Entstehung des sogenannten Moivreschen Satzes" [On the history of the origin of the so-called Moivre theorem]. Bibliotheca Mathematica. 3rd series (in German). 2: 97–102.; see p. 98.

[2] Lial, Margaret L.; Hornsby, John; Schneider, David I.; Callie J., Daniels (2008). College Algebra and Trigonometry (4th ed.). Boston: Pearson/Addison Wesley. p. 792. ISBN 9780321497444.

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De Moivre's formula

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