Fungrim entry: 20828c

\frac{d}{d a}\, \operatorname{agm}\!\left(a, b\right) = \frac{\operatorname{agm}\!\left(a, b\right)}{\pi a \left(a - b\right)} \left(\pi a - 2 \operatorname{agm}\!\left(a, b\right) E\!\left({\left(\frac{a - b}{a + b}\right)}^{2}\right)\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; a \ne b \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

References:

http://functions.wolfram.com/09.54.20.0001.01

TeX:

\frac{d}{d a}\, \operatorname{agm}\!\left(a, b\right) = \frac{\operatorname{agm}\!\left(a, b\right)}{\pi a \left(a - b\right)} \left(\pi a - 2 \operatorname{agm}\!\left(a, b\right) E\!\left({\left(\frac{a - b}{a + b}\right)}^{2}\right)\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; a \ne b \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

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
`Pi`	$\pi$	The constant pi (3.14...)
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("20828c"),
    Formula(Equal(ComplexDerivative(AGM(a, b), For(a, a)), Mul(Div(AGM(a, b), Mul(Mul(Pi, a), Sub(a, b))), Sub(Mul(Pi, a), Mul(Mul(2, AGM(a, b)), EllipticE(Pow(Div(Sub(a, b), Add(a, b)), 2))))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotEqual(a, b), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 0)))),
    References("http://functions.wolfram.com/09.54.20.0001.01"))

Topics using this entry

Arithmetic-geometric mean