Fungrim entry: 71a264

$\sin^{2 n + 1}\!\left(z\right) = \frac{1}{{4}^{n}} \sum_{k=0}^{n} {\left(-1\right)}^{n + k} {2 n + 1 \choose k} \sin\!\left(\left(2 n - 2 k + 1\right) z\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
TeX:
\sin^{2 n + 1}\!\left(z\right) = \frac{1}{{4}^{n}} \sum_{k=0}^{n} {\left(-1\right)}^{n + k} {2 n + 1 \choose k} \sin\!\left(\left(2 n - 2 k + 1\right) z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Pow${a}^{b}$ Power
Sin$\sin(z)$ Sine
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("71a264"),
Formula(Equal(Pow(Sin(z), Add(Mul(2, n), 1)), Mul(Div(1, Pow(4, n)), Sum(Mul(Mul(Pow(-1, Add(n, k)), Binomial(Add(Mul(2, n), 1), k)), Sin(Mul(Add(Sub(Mul(2, n), Mul(2, k)), 1), z))), For(k, 0, n))))),
Variables(z, n),
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