# 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

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

2019-12-30 15:00:46.909060 UTC