Fungrim entry: 447541

\left[ \frac{d^{n}}{{d x}^{n}} \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = \frac{{\left(-1\right)}^{n} n !}{{8}^{n}} \text{A060691}\!\left(n\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0}

References:

Sequence A060691 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

\left[ \frac{d^{n}}{{d x}^{n}} \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = \frac{{\left(-1\right)}^{n} n !}{{8}^{n}} \text{A060691}\!\left(n\right)

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("447541"),
    Formula(Equal(ComplexDerivative(AGM(1, x), For(x, 1, n)), Mul(Div(Mul(Pow(-1, n), Factorial(n)), Pow(8, n)), SloaneA("060691", n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

Arithmetic-geometric mean