Fungrim entry: 3b175b

\left|P''_{n}(x)\right| \le \frac{{2}^{5 / 2}}{\sqrt{\pi}} \frac{{n}^{3 / 2}}{{\left(1 - {x}^{2}\right)}^{5 / 4}}

Assumptions:

n \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
`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("3b175b"),
    Formula(LessEqual(Abs(ComplexDerivative(LegendrePolynomial(n, x), For(x, x, 2))), Mul(Div(Pow(2, Div(5, 2)), Sqrt(Pi)), 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))))

Topics using this entry

Legendre polynomials