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Fungrim entry: 6476bd

Pn(r)(x)2r+1/2πnr1/2(1x2)(2n+1)/4\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:nZ0  and  rZ0  and  1<x<1n \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
Absz\left|z\right| Absolute value
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
Piπ\pi The constant pi (3.14...)
ZZGreaterEqualZn\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))))

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2020-04-08 16:14:44.404316 UTC