De Moivre's formula

Moivre's theorem, also called de Moivre's theorem or de Moivre's formula, states that for any complex number (and thus any real number) xand any natural number nthe relation

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

applies.

It is named in honor of Abraham de Moivre, who found this theorem in the first decade of the 18th century. De Moivre himself had the formula, according to his own statement, from his teacher Isaac Newton and used it in various of his writings, even though he never explicitly wrote it down (this was done only by Leonhard Euler 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 \cos x+i\,\sin xcan also be abbreviated as \operatorname {cis}\,x.

Derivation

Moivre's theorem can be calculated 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}}

can 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

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

then

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

a multivalued function, but not

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

This means that

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


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