Fungrim entry: 61d8f3

\left(1 + {z}^{2}\right) y''(z) + 2 z y'(z) = 0\; \text{ where } y(z) = {c}_{1} + {c}_{2} \operatorname{atan}(z)

Assumptions:

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
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	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))))))

Topics using this entry

Inverse tangent