Fungrim entry: d0505f

{\left(\cos\!\left(z\right) + i \sin\!\left(z\right)\right)}^{n} = \cos\!\left(n z\right) + i \sin\!\left(n z\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

{\left(\cos\!\left(z\right) + i \sin\!\left(z\right)\right)}^{n} = \cos\!\left(n z\right) + i \sin\!\left(n z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Sin`	$\sin\!\left(z\right)$	Sine
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d0505f"),
    Formula(Equal(Pow(Add(Cos(z), Mul(ConstI, Sin(z))), n), Add(Cos(Mul(n, z)), Mul(ConstI, Sin(Mul(n, z)))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZ))))

Topics using this entry

Sine

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC