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Fungrim entry: 61d8f3

(1+z2)y(z)+2zy(z)=0   where y(z)=c1+c2atan(z)\left(1 + {z}^{2}\right) y''(z) + 2 z y'(z) = 0\; \text{ where } y(z) = {c}_{1} + {c}_{2} \operatorname{atan}(z)
Assumptions:zCandc1Candc2Candiz(,1][1,)z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{2} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, i z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)
TeX:
\left(1 + {z}^{2}\right) y''(z) + 2 z y'(z) = 0\; \text{ where } y(z) = {c}_{1} + {c}_{2} \operatorname{atan}(z)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {c}_{2} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, i z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Atanatan(z)\operatorname{atan}(z) Inverse tangent
CCC\mathbb{C} Complex numbers
ConstIii Imaginary unit
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Source code for this entry:
Entry(ID("61d8f3"),
    Formula(Where(Equal(Add(Mul(Add(1, Pow(z, 2)), ComplexDerivative(y(z), For(z, z, 2))), Mul(Mul(2, z), ComplexDerivative(y(z), For(z, z, 1)))), 0), Equal(y(z), Add(Subscript(c, 1), Mul(Subscript(c, 2), Atan(z)))))),
    Variables(z, Subscript(c, 1), Subscript(c, 2)),
    Assumptions(And(Element(z, CC), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), NotElement(Mul(ConstI, z), Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-01-31 18:09:28.494564 UTC