Fungrim entry: eca4ce

\operatorname{atan2}\!\left(y, x\right) = \operatorname{Im}\!\left(\log\!\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) = \operatorname{Im}\!\left(\log\!\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
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("eca4ce"),
    Formula(Equal(Atan2(y, x), Im(Log(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