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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"))

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2021-03-15 19:12:00.328586 UTC