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Fungrim entry: 155343

Pn ⁣(x)2I0 ⁣(2nx12)2e2nx1/2\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:nZ0andxRn \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
Absz\left|z\right| Absolute value
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
BesselIIν ⁣(z)I_{\nu}\!\left(z\right) Modified Bessel function of the first kind
Sqrtz\sqrt{z} Principal square root
Expez{e}^{z} Exponential function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
RRR\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))))

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2019-08-21 11:44:15.926409 UTC