Fungrim entry: 155343

\left|P_{n}\!\left(x\right)\right| \le 2 I_{0}\!\left(2 n \sqrt{\frac{\left|x - 1\right|}{2}}\right) \le 2 {e}^{2 n \sqrt{\left|x - 1\right| / 2}}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}

TeX:

\left|P_{n}\!\left(x\right)\right| \le 2 I_{0}\!\left(2 n \sqrt{\frac{\left|x - 1\right|}{2}}\right) \le 2 {e}^{2 n \sqrt{\left|x - 1\right| / 2}}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Exp`	${e}^{z}$	Exponential function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("155343"),
    Formula(LessEqual(Abs(LegendrePolynomial(n, x)), Mul(2, BesselI(0, Mul(Mul(2, n), Sqrt(Div(Abs(Sub(x, 1)), 2))))), Mul(2, Exp(Mul(Mul(2, n), Sqrt(Div(Abs(Sub(x, 1)), 2))))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, RR))))

Topics using this entry

Legendre polynomials