# Fungrim entry: 83abff

$\frac{d^{r}}{{d z}^{r}} \sqrt{z} = {\left(-1\right)}^{r} \left(-\frac{1}{2}\right)_{r} {z}^{r - 1 / 2}$
Assumptions:$z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$
TeX:
\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}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
RisingFactorial$\left(z\right)_{k}$ Rising factorial
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("83abff"),
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)))))

## 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