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Fungrim entry: 3b175b

Pn(x)25/2πn3/2(1x2)5/4\left|P''_{n}(x)\right| \le \frac{{2}^{5 / 2}}{\sqrt{\pi}} \frac{{n}^{3 / 2}}{{\left(1 - {x}^{2}\right)}^{5 / 4}}
Assumptions:nZ0and1<x<1n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, -1 < x < 1
TeX:
\left|P''_{n}(x)\right| \le \frac{{2}^{5 / 2}}{\sqrt{\pi}} \frac{{n}^{3 / 2}}{{\left(1 - {x}^{2}\right)}^{5 / 4}}

n \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
ConstPiπ\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("3b175b"),
    Formula(LessEqual(Abs(ComplexDerivative(LegendrePolynomial(n, x), For(x, x, 2))), Mul(Div(Pow(2, Div(5, 2)), Sqrt(ConstPi)), Div(Pow(n, Div(3, 2)), Pow(Sub(1, Pow(x, 2)), Div(5, 4)))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Less(-1, x, 1))))

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2019-10-05 13:11:19.856591 UTC