Fungrim entry: 073e1a

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

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\operatorname{Re}\!\left(z\right) \lt 0 \,\mathbin{\operatorname{or}}\, \left(\operatorname{Re}\!\left(z\right) = 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \in \left(-\infty, -1\right) \cup \left(0, 1\right)\right)\right)

TeX:

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

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\operatorname{Re}\!\left(z\right) \lt 0 \,\mathbin{\operatorname{or}}\, \left(\operatorname{Re}\!\left(z\right) = 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \in \left(-\infty, -1\right) \cup \left(0, 1\right)\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`Atan`	$\operatorname{atan}\!\left(z\right)$	Inverse tangent
`ConstPi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("073e1a"),
    Formula(Equal(Atan(Div(1, z)), Sub(Neg(Div(ConstPi, 2)), Atan(z)))),
    Variables(z),
    Assumptions(And(Element(z, CC), Or(Less(Re(z), 0), And(Equal(Re(z), 0), Element(Im(z), Union(OpenInterval(Neg(Infinity), -1), OpenInterval(0, 1))))))))

Topics using this entry

Inverse tangent