Fungrim entry: b65d19

\operatorname{Im}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{4} \log\!\left(\frac{{x}^{2} + {\left(1 + y\right)}^{2}}{{x}^{2} + {\left(1 - y\right)}^{2}}\right)

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x + y i \notin \left\{-i, i\right\}

TeX:

\operatorname{Im}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{4} \log\!\left(\frac{{x}^{2} + {\left(1 + y\right)}^{2}}{{x}^{2} + {\left(1 - y\right)}^{2}}\right)

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x + y i \notin \left\{-i, i\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`Atan`	$\operatorname{atan}\!\left(z\right)$	Inverse tangent
`ConstI`	$i$	Imaginary unit
`Log`	$\log\!\left(z\right)$	Natural logarithm
`Pow`	${a}^{b}$	Power
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("b65d19"),
    Formula(Equal(Im(Atan(Add(x, Mul(y, ConstI)))), Mul(Div(1, 4), Log(Div(Add(Pow(x, 2), Pow(Add(1, y), 2)), Add(Pow(x, 2), Pow(Sub(1, y), 2))))))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR), NotElement(Add(x, Mul(y, ConstI)), Set(Neg(ConstI), ConstI)))))

Topics using this entry

Inverse tangent