Fungrim entry: 36171f

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

Assumptions:

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

TeX:

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

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; 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
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`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("36171f"),
    Formula(Equal(ComplexDerivative(Atan(z), For(z, z, n)), Mul(Div(Mul(Pow(-1, n), Factorial(Sub(n, 1))), Mul(2, ConstI)), Sub(Div(1, Pow(Add(z, ConstI), n)), Div(1, Pow(Sub(z, ConstI), n)))))),
    Variables(z, n),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(z, CC), NotElement(Mul(ConstI, z), Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

Topics using this entry

Inverse tangent