Fungrim home page

Fungrim entry: 36171f

atan(n)(z)=(1)n(n1)!2i(1(z+i)n1(zi)n){\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: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)
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
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Atanatan(z)\operatorname{atan}(z) Inverse tangent
Powab{a}^{b} Power
Factorialn!n ! Factorial
ConstIii Imaginary unit
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
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("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.

2019-11-11 15:50:15.016492 UTC