Fungrim entry: a4cc5a

2 a \left({b}^{2} - {a}^{2}\right) {\left(f'(a)\right)}^{2} - a {\left(f(a)\right)}^{2} + \left(\left(3 {a}^{2} - {b}^{2}\right) f'(a) + a \left({a}^{2} - {b}^{2}\right) f''(a)\right) f(a) = 0\; \text{ where } f(a) = \operatorname{agm}\!\left(a, b\right)

Assumptions:

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

References:

http://functions.wolfram.com/09.54.13.0001.01

TeX:

2 a \left({b}^{2} - {a}^{2}\right) {\left(f'(a)\right)}^{2} - a {\left(f(a)\right)}^{2} + \left(\left(3 {a}^{2} - {b}^{2}\right) f'(a) + a \left({a}^{2} - {b}^{2}\right) f''(a)\right) f(a) = 0\; \text{ where } f(a) = \operatorname{agm}\!\left(a, b\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("a4cc5a"),
    Formula(Where(Equal(Add(Sub(Mul(Mul(Mul(2, a), Sub(Pow(b, 2), Pow(a, 2))), Pow(ComplexDerivative(f(a), For(a, a)), 2)), Mul(a, Pow(f(a), 2))), Mul(Add(Mul(Sub(Mul(3, Pow(a, 2)), Pow(b, 2)), ComplexDerivative(f(a), For(a, a))), Mul(Mul(a, Sub(Pow(a, 2), Pow(b, 2))), ComplexDerivative(f(a), For(a, a, 2)))), f(a))), 0), Def(f(a), AGM(a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 0)))),
    References("http://functions.wolfram.com/09.54.13.0001.01"))

Topics using this entry

Arithmetic-geometric mean