# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC