Richard J. Mathar

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math.CA2025

Chebyshev Expansion of the Zernike Polynomials

Richard J. Mathar

The even and odd Zernike Polynomials R_n^m(x) can be expanded into sums of even and odd Chebyshev Polynomials T_i(x). This manuscript provides closed forms for the rational expansi…

math.CA2024

Chebyshev approximation of $x^{m} (- lo g x)^{l}$ in the interval $0 \leq x \leq 1$

Richard J. Mathar

The series expansion of $x^{m} (- lo g x)^{l}$ 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…

math.CA2024

Yet Another Table of Integrals

Richard J. Mathar

This collection of sums and integrals has been harvested from the mathematical and physical literature in unstructured ways. Its main use is backtracking the original sources whene…

Chebyshev Expansion of the Zernike Polynomials

Chebyshev approximation of xm(−logx)l in the interval 0≤x≤1

Yet Another Table of Integrals

Chebyshev approximation of $x^{m} (- lo g x)^{l}$ in the interval $0 \leq x \leq 1$