Fungrim entry: 12765e

\operatorname{atan}\!\left(z\right) = -\frac{i}{2} \log\!\left(\frac{1 + i z}{1 - i z}\right)

Assumptions:

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

TeX:

\operatorname{atan}\!\left(z\right) = -\frac{i}{2} \log\!\left(\frac{1 + i z}{1 - i z}\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Atan`	$\operatorname{atan}\!\left(z\right)$	Inverse tangent
`ConstI`	$i$	Imaginary unit
`Log`	$\log\!\left(z\right)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("12765e"),
    Formula(Equal(Atan(z), Mul(Neg(Div(ConstI, 2)), Log(Div(Add(1, Mul(ConstI, z)), Sub(1, Mul(ConstI, z))))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(Mul(z, ConstI), ClosedOpenInterval(1, Infinity)))))

Topics using this entry

Inverse tangent