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

Square roots