Fungrim entry: 9dec3e

\operatorname{atan2}\!\left(y, x\right) = -i \log\!\left(\operatorname{sgn}\!\left(x + y i\right)\right)

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x + y i \ne 0

TeX:

\operatorname{atan2}\!\left(y, x\right) = -i \log\!\left(\operatorname{sgn}\!\left(x + y i\right)\right)

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x + y i \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Atan2`	$\operatorname{atan2}\!\left(y, x\right)$	Two-argument inverse tangent
`ConstI`	$i$	Imaginary unit
`Log`	$\log\!\left(z\right)$	Natural logarithm
`Sign`	$\operatorname{sgn}\!\left(z\right)$	Sign function
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("9dec3e"),
    Formula(Equal(Atan2(y, x), Mul(Neg(ConstI), Log(Sign(Add(x, Mul(y, ConstI))))))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR), Unequal(Add(x, Mul(y, ConstI)), 0))))

Topics using this entry

Inverse tangent