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

TL;DR

This paper develops a method for approximating the function $x^m (- ext{log} x)^l$ using Chebyshev polynomials, reducing complex integrals to known sums involving polygamma functions.

Contribution

It introduces a reduction technique for Chebyshev series expansion integrals of $x^m (- ext{log} x)^l$, linking them to polygamma function sums.

Findings

Integral evaluation simplified via partial integration.

Expresses integrals in terms of finite sums over polygamma functions.

Provides a practical approach for Chebyshev approximation of complex functions.

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.