Chebyshev approximation of $x^m (-\log x)^l$ in the interval $0\le x \le 1$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912003745906688 |
|---|---|
| author | Mathar, Richard J. |
| author_facet | Mathar, Richard J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15212 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chebyshev approximation of $x^m (-\log x)^l$ in the interval $0\le x \le 1$ Mathar, Richard J. Classical Analysis and ODEs Primary 26A09, Secondary 41A10 G.1.2 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. |
| title | Chebyshev approximation of $x^m (-\log x)^l$ in the interval $0\le x \le 1$ |
| topic | Classical Analysis and ODEs Primary 26A09, Secondary 41A10 G.1.2 |
| url | https://arxiv.org/abs/2408.15212 |