# Fungrim entry: 61d8f3

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

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
Derivative$\frac{d}{d z}\, f\!\left(z\right)$ Derivative
Atan$\operatorname{atan}\!\left(z\right)$ 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)), Derivative(y(z), Tuple(z, z, 2))), Mul(Mul(2, z), Derivative(y(z), Tuple(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.

2019-09-16 21:17:18.797188 UTC