Fungrim entry: 90631b

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

References:

M. A. Boutiche and M. Rahmani (2017), On the higher derivatives of the inverse tangent function, https://arxiv.org/abs/1712.03521, Theorem 9

TeX:

{\operatorname{atan}}^{(n)}(z) = \frac{\left(n - 1\right)!}{{\left(1 + {z}^{2}\right)}^{\left( n + 1 \right) / 2}} U_{n - 1}\!\left(-\frac{z}{\sqrt{1 + {z}^{2}}}\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
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`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("90631b"),
    Formula(Equal(ComplexDerivative(Atan(z), For(z, z, n)), Mul(Div(Factorial(Sub(n, 1)), Pow(Add(1, Pow(z, 2)), Div(Add(n, 1), 2))), ChebyshevU(Sub(n, 1), Neg(Div(z, Sqrt(Add(1, Pow(z, 2))))))))),
    Variables(z, n),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(z, CC), NotElement(Mul(ConstI, z), Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))),
    References("M. A. Boutiche and M. Rahmani (2017), On the higher derivatives of the inverse tangent function, https://arxiv.org/abs/1712.03521, Theorem 9"))

Topics using this entry

Inverse tangent