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

Sine