# 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

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