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Fungrim entry: d0505f

(cos⁡(z)+isin⁡(z))n=cos⁡ ⁣(nz)+isin⁡ ⁣(nz){\left(\cos(z) + i \sin(z)\right)}^{n} = \cos\!\left(n z\right) + i \sin\!\left(n z\right)
Assumptions:z∈C  and⁡  n∈Zz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}
TeX:
{\left(\cos(z) + i \sin(z)\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
Powab{a}^{b} Power
Coscos⁡(z)\cos(z) Cosine
ConstIii Imaginary unit
Sinsin⁡(z)\sin(z) Sine
CCC\mathbb{C} Complex numbers
ZZZ\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

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

2021-03-15 19:12:00.328586 UTC