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Fungrim entry: e3f8a4

sin ⁣(nz)=k=0(n1)/2(1)k(n2k+1)cosn2k1 ⁣(z)sin2k+1 ⁣(z)\sin\!\left(n z\right) = \sum_{k=0}^{\left\lfloor \left( n - 1 \right) / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose 2 k + 1} \cos^{n - 2 k - 1}\!\left(z\right) \sin^{2 k + 1}\!\left(z\right)
Assumptions:zCandnZ0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0}
TeX:
\sin\!\left(n z\right) = \sum_{k=0}^{\left\lfloor \left( n - 1 \right) / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose 2 k + 1} \cos^{n - 2 k - 1}\!\left(z\right) \sin^{2 k + 1}\!\left(z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Sinsin(z)\sin(z) Sine
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Binomial(nk){n \choose k} Binomial coefficient
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("e3f8a4"),
    Formula(Equal(Sin(Mul(n, z)), Sum(Mul(Mul(Mul(Pow(-1, k), Binomial(n, Add(Mul(2, k), 1))), Pow(Cos(z), Sub(Sub(n, Mul(2, k)), 1))), Pow(Sin(z), Add(Mul(2, k), 1))), For(k, 0, Floor(Div(Sub(n, 1), 2)))))),
    Variables(n, z),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(0)))))

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

2019-10-05 13:11:19.856591 UTC