# Fungrim entry: 503d4d

$\operatorname{atan}(x) - \operatorname{atan}(y) = \operatorname{atan}\!\left(\frac{x - y}{1 + x y}\right)$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1 \;\mathbin{\operatorname{and}}\; \left|y\right| < 1$
Alternative assumptions:$x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x y > -1$
TeX:
\operatorname{atan}(x) - \operatorname{atan}(y) = \operatorname{atan}\!\left(\frac{x - y}{1 + x y}\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1 \;\mathbin{\operatorname{and}}\; \left|y\right| < 1

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x y > -1
Definitions:
Fungrim symbol Notation Short description
Atan$\operatorname{atan}(z)$ Inverse tangent
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("503d4d"),
Formula(Equal(Sub(Atan(x), Atan(y)), Atan(Div(Sub(x, y), Add(1, Mul(x, y)))))),
Variables(x, y),
Assumptions(And(Element(x, CC), Element(y, CC), Less(Abs(x), 1), Less(Abs(y), 1)), And(Element(x, RR), Element(y, RR), Greater(Mul(x, y), -1))))

## 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