Fungrim home page

Fungrim entry: 90631b

atan(n)(z)=(n1)!(1+z2)(n+1)/2Un1 ⁣(z1+z2){\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:nZ1andzCandiz(,1][1,)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
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Atanatan(z)\operatorname{atan}(z) Inverse tangent
Factorialn!n ! Factorial
Powab{a}^{b} Power
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Sqrtz\sqrt{z} Principal square root
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ConstIii Imaginary unit
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ClosedOpenInterval[a,b)\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.

2019-10-05 13:11:19.856591 UTC