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Fungrim entry: d0505f

(cos ⁣(z)+isin ⁣(z))n=cos ⁣(nz)+isin ⁣(nz){\left(\cos\!\left(z\right) + i \sin\!\left(z\right)\right)}^{n} = \cos\!\left(n z\right) + i \sin\!\left(n z\right)
Assumptions:zCandnZz \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}
{\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}
Fungrim symbol Notation Short description
Powab{a}^{b} Power
ConstIii Imaginary unit
Sinsin ⁣(z)\sin\!\left(z\right) Sine
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
Source code for this entry:
    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))))

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

2019-08-25 15:30:03.056001 UTC