Fungrim entry: bfc13f

\operatorname{atan}\!\left(\frac{1}{z}\right) = \frac{\pi}{2} \operatorname{csgn}\!\left(\frac{1}{z}\right) - \operatorname{atan}(z)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; i z \notin \left\{0\right\} \cup \left(-\infty, -1\right] \cup \left[1, \infty\right)

TeX:

\operatorname{atan}\!\left(\frac{1}{z}\right) = \frac{\pi}{2} \operatorname{csgn}\!\left(\frac{1}{z}\right) - \operatorname{atan}(z)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; i z \notin \left\{0\right\} \cup \left(-\infty, -1\right] \cup \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Pi`	$\pi$	The constant pi (3.14...)
`Csgn`	$\operatorname{csgn}(z)$	Real-valued sign function for complex numbers
`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("bfc13f"),
    Formula(Equal(Atan(Div(1, z)), Sub(Mul(Div(Pi, 2), Csgn(Div(1, z))), Atan(z)))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(Mul(ConstI, z), Union(Set(0), OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

Topics using this entry

Inverse tangent