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Fungrim entry: 83abff

drdzrz=(1)r(12)rzr1/2\frac{d^{r}}{{d z}^{r}} \sqrt{z} = {\left(-1\right)}^{r} \left(-\frac{1}{2}\right)_{r} {z}^{r - 1 / 2}
Assumptions:zC(,0]  and  rZ0z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}
\frac{d^{r}}{{d z}^{r}} \sqrt{z} = {\left(-1\right)}^{r} \left(-\frac{1}{2}\right)_{r} {z}^{r - 1 / 2}

z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(ComplexDerivative(Sqrt(z), For(z, z, r)), Mul(Mul(Pow(-1, r), RisingFactorial(Neg(Div(1, 2)), r)), Pow(z, Sub(r, Div(1, 2)))))),
    Variables(z, r),
    Assumptions(And(Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Element(r, ZZGreaterEqual(0)))))

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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