Fungrim entry: 6476bd

\left|\frac{d^{r}}{{d x}^{r}} P_{n}\!\left(x\right)\right| \le \frac{{2}^{r + 1 / 2}}{\sqrt{\pi}} \frac{{n}^{r - 1 / 2}}{{\left(1 - {x}^{2}\right)}^{\left( 2 n + 1 \right) / 4}}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, -1 \lt x \lt 1

TeX:

\left|\frac{d^{r}}{{d x}^{r}} P_{n}\!\left(x\right)\right| \le \frac{{2}^{r + 1 / 2}}{\sqrt{\pi}} \frac{{n}^{r - 1 / 2}}{{\left(1 - {x}^{2}\right)}^{\left( 2 n + 1 \right) / 4}}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, -1 \lt x \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("6476bd"),
    Formula(LessEqual(Abs(Derivative(LegendrePolynomial(n, x), Tuple(x, x, r))), Mul(Div(Pow(2, Add(r, Div(1, 2))), Sqrt(ConstPi)), Div(Pow(n, Sub(r, Div(1, 2))), Pow(Sub(1, Pow(x, 2)), Div(Add(Mul(2, n), 1), 4)))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(r, ZZGreaterEqual(0)), Less(-1, x, 1))))

Topics using this entry

Legendre polynomials