Fungrim home page

Fungrim entry: a4eb86

atan(z)=2z(1+z2)2\operatorname{atan}''(z) = -\frac{2 z}{{\left(1 + {z}^{2}\right)}^{2}}
Assumptions:zC  and  iz(,1][1,)z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; i z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)
\operatorname{atan}''(z) = -\frac{2 z}{{\left(1 + {z}^{2}\right)}^{2}}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; i z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)
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
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:
    Formula(Equal(ComplexDerivative(Atan(z), For(z, z, 2)), Neg(Div(Mul(2, z), Pow(Add(1, Pow(z, 2)), 2))))),
    Assumptions(And(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