Fungrim entry: a4eb86

\operatorname{atan}''(z) = -\frac{2 z}{{\left(1 + {z}^{2}\right)}^{2}}

Assumptions:

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

TeX:

\operatorname{atan}''(z) = -\frac{2 z}{{\left(1 + {z}^{2}\right)}^{2}}

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

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Atan`	$\operatorname{atan}\!\left(z\right)$	Inverse tangent
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("a4eb86"),
    Formula(Equal(Derivative(Atan(z), Tuple(z, z, 2)), Neg(Div(Mul(2, z), Pow(Add(1, Pow(z, 2)), 2))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(Mul(ConstI, z), Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

Topics using this entry

Inverse tangent