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Fungrim entry: a4cc5a

2a(b2a2)(f(a))2a(f(a))2+((3a2b2)f(a)+a(a2b2)f(a))f(a)=0   where f(a)=agm ⁣(a,b)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:aC  and  bC  and  b0  and  ab(,0]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]
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]
Fungrim symbol Notation Short description
Powab{a}^{b} Power
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
Source code for this entry:
    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)))),

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2021-03-15 19:12:00.328586 UTC