Fungrim entry: 7954ad

\operatorname{atan}\!\left(z\right) = \operatorname{asin}\!\left(\frac{z}{\sqrt{1 + {z}^{2}}}\right)

Assumptions:

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

TeX:

\operatorname{atan}\!\left(z\right) = \operatorname{asin}\!\left(\frac{z}{\sqrt{1 + {z}^{2}}}\right)

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`	$\operatorname{atan}\!\left(z\right)$	Inverse tangent
`Sqrt`	$\sqrt{z}$	Principal square root
`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("7954ad"),
    Formula(Equal(Atan(z), Asin(Div(z, Sqrt(Add(1, Pow(z, 2))))))),
    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