# Fungrim entry: 27b2bb

$\sum_{n=1}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n} = -\frac{1}{2} \log\!\left(1 - 2 x z + {z}^{2}\right)$
Assumptions:$x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1$
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
Sum$\sum_{n} f(n)$ Sum
ChebyshevT$T_{n}\!\left(x\right)$ Chebyshev polynomial of the first kind
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
Log$\log(z)$ Natural logarithm
ClosedInterval$\left[a, b\right]$ Closed interval
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("27b2bb"),
Formula(Equal(Sum(Mul(ChebyshevT(n, x), Div(Pow(z, n), n)), For(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))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC