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Fungrim entry: 27b2bb

n=1Tn ⁣(x)znn=12log ⁣(12xz+z2)\sum_{n=1}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n} = -\frac{1}{2} \log\!\left(1 - 2 x z + {z}^{2}\right)
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=1}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n} = -\frac{1}{2} \log\!\left(1 - 2 x z + {z}^{2}\right)

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
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
Powab{a}^{b} Power
Infinity\infty Positive infinity
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
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("27b2bb"),
    Formula(Equal(Sum(Mul(ChebyshevT(n, x), Div(Pow(z, n), n)), Tuple(n, 1, Infinity)), Mul(Neg(Div(1, 2)), Log(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))))

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

2019-09-20 18:07:53.062439 UTC