# 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

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

2021-03-15 19:12:00.328586 UTC