# Fungrim entry: 6476bd

$\left|{P}^{(r)}_{n}(x)\right| \le \frac{{2}^{r + 1 / 2}}{\sqrt{\pi}} \frac{{n}^{r - 1 / 2}}{{\left(1 - {x}^{2}\right)}^{\left( 2 n + 1 \right) / 4}}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; -1 < x < 1$
TeX:
\left|{P}^{(r)}_{n}(x)\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 < x < 1
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
LegendrePolynomial$P_{n}\!\left(z\right)$ Legendre polynomial
Pow${a}^{b}$ Power
Sqrt$\sqrt{z}$ Principal square root
Pi$\pi$ 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(ComplexDerivative(LegendrePolynomial(n, x), For(x, x, r))), Mul(Div(Pow(2, Add(r, Div(1, 2))), Sqrt(Pi)), 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

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

2020-04-08 16:14:44.404316 UTC