Fungrim entry: e3f8a4

$\sin\!\left(n z\right) = \sum_{k=0}^{\left\lfloor \left( n - 1 \right) / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose 2 k + 1} \cos^{n - 2 k - 1}\!\left(z\right) \sin^{2 k + 1}\!\left(z\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
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
Sin$\sin(z)$ Sine
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Binomial${n \choose k}$ Binomial coefficient
Cos$\cos(z)$ Cosine
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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)))))

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