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Fungrim entry: 8fbf69

atan(z)=11+z2\operatorname{atan}'(z) = \frac{1}{1 + {z}^{2}}
Assumptions:zCandiz(,1][1,)z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, i z \notin \left(-\infty, -1\right] \cup \left[1, \infty\right)
TeX:
\operatorname{atan}'(z) = \frac{1}{1 + {z}^{2}}

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
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("8fbf69"),
    Formula(Equal(ComplexDerivative(Atan(z), For(z, z, 1)), Div(1, Add(1, Pow(z, 2))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(Mul(ConstI, z), Union(OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(1, Infinity))))))

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

2020-01-31 18:09:28.494564 UTC