Moivrescher's sentence

The Moivresche theorem , also called de Moivre's theorem or de Moivre's formula , states that for every complex number (and thus every real number ) and every natural number there is a connection ${\displaystyle x}$ ${\displaystyle n}$

${\displaystyle \left(\cos x+i\,\sin x\right)^{n}=\cos \left(n\,x\right)+i\,\sin \left(n\,x\right)}$

applies.

It bears his name in honor of Abraham de Moivre , who found this sentence in the first decade of the 18th century. De Moivre himself said he had the formula from his teacher Isaac Newton and used it in various of his writings, even if he never wrote it down explicitly ( Leonhard Euler did that in 1748, Introductio in analysin infinitorum , where he also established Euler's formula ) .

The formula connects the complex numbers with trigonometry so that the complex numbers can be represented trigonometrically. The expression can also be shortened as . ${\displaystyle \cos x+i\,\sin x}$ ${\displaystyle \operatorname {cis} \,x}$

Derivation

Moivresche's theorem can be used with Euler's formula

${\displaystyle e^{\mathrm {i} x}=\cos x+\mathrm {i} \sin x}$

the complex exponential function and its functional equation

${\displaystyle \left(e^{\mathrm {i} x}\right)^{n}=e^{\mathrm {i} xn}}$

be derived.

An alternative proof results from the product representation (see addition theorems )

${\displaystyle (\cos \varphi +\mathrm {i} \sin \varphi )\cdot (\cos \psi +\mathrm {i} \sin \psi )=\cos(\varphi +\psi )+\mathrm {i} \sin(\varphi +\psi )}$

by complete induction .

generalization

If

${\displaystyle z,w\in \mathbb {C} }$

then

${\displaystyle \left(\cos z+i\,\sin z\right)^{w}}$

a multi-valued function , but not

${\displaystyle \cos \left(w\,z\right)+i\,\sin \left(w\,z\right).}$

This applies

${\displaystyle \cos \left(w\,z\right)+i\,\sin \left(w\,z\right)\in \{\left(\cos z+i\,\sin z\right)^{w}\}.}$

See also

• Root of unity

literature

• Anton von Braunmühl : Lectures on the history of trigonometry . History of trigonometry. Contains: Part 1 - From the oldest times to the invention of logarithms, Part 2 From the invention of logarithms to the present day. Reprographic reprint of the 1st edition. M. Sendet, Niederwalluf near Wiesbaden 1971, ISBN 3-500-23250-7 (first edition by Teubner, Leipzig, 1900-1903).
• Hans Kerner, Wolf von Wahl: Mathematics for Physicists . 2nd revised and expanded edition. Springer, Berlin / Heidelberg / New York 2007, ISBN 978-3-540-72479-7 .

Individual evidence

1. ↑ Kerner and Wahl (2007), p. 70
2. ↑ Braunmühl (1971), Part 2, p. 75
3. ↑ Braunmühl (1971), Part 2, p. 78
4. ^ Nahin, An imaginary tale, Princeton University Press 1998, p. 56