# 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

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

2021-03-15 19:12:00.328586 UTC