Fungrim entry: 90a864

\operatorname{atan}(z) = \int_{0}^{z} \frac{1}{1 + {t}^{2}} \, dt

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; i z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)

TeX:

\operatorname{atan}(z) = \int_{0}^{z} \frac{1}{1 + {t}^{2}} \, dt

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; i z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`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("90a864"),
    Formula(Equal(Atan(z), Integral(Div(1, Add(1, Pow(t, 2))), For(t, 0, z)))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(Mul(ConstI, z), Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

Topics using this entry

Inverse tangent