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

Abstract

The series expansion of $x^m(-\log x{)}^l$ in terms of the shifted Chebyshev Polynomials $T_n^{\ast }(x)$ 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.

9 pages, no figures. Integrals table in the anc directory