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Fungrim entry: d84519

n=0Pn ⁣(x)zn=112xz+z2\sum_{n=0}^{\infty} P_{n}\!\left(x\right) {z}^{n} = \frac{1}{\sqrt{1 - 2 x z + {z}^{2}}}
Assumptions:x[1,1]andzCandz<1x \in \left[-1, 1\right] \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| < 1
TeX:
\sum_{n=0}^{\infty} P_{n}\!\left(x\right) {z}^{n} = \frac{1}{\sqrt{1 - 2 x z + {z}^{2}}}

x \in \left[-1, 1\right] \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| < 1
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
Powab{a}^{b} Power
Infinity\infty Positive infinity
Sqrtz\sqrt{z} Principal square root
ClosedInterval[a,b]\left[a, b\right] Closed interval
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Source code for this entry:
Entry(ID("d84519"),
    Formula(Equal(Sum(Mul(LegendrePolynomial(n, x), Pow(z, n)), For(n, 0, Infinity)), Div(1, Sqrt(Add(Sub(1, Mul(Mul(2, x), z)), Pow(z, 2)))))),
    Variables(x, z),
    Assumptions(And(Element(x, ClosedInterval(-1, 1)), Element(z, CC), Less(Abs(z), 1))))

Topics using this entry

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

2019-09-22 15:43:45.410764 UTC