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arXiv · 2408.15212

Chebyshev approximation of $x^m (-\log x)^l$ in the interval $0\le x \le 1$

Richard J. Mathar
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Abstract

The series expansion of $x^m (-\log x)^l$ in terms of the shifted Chebyshev Polynomials $T_n^*(x)$ requires evaluation of the integral family $\int_0^1 x^m (-\log x)^l dx / \sqrt{x-x^2}$. We demonstrate that these can be reduced by partial integration to sums over integrals with exponent $m=0$ which have known representations as finite sums over polygamma functions.

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Richard J. Mathar. 2024-08-27. Chebyshev approximation of $x^m (-\log x)^l$ in the interval $0\le x \le 1$. https://arxiv.org/abs/2408.15212

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