Three algorithms for Hadamard finite-part integrals and fractional derivatives

Abstract Three algorithms for the evaluation of the Hadamard finite-part integral of the form ∫ 1 -1 ( f(t) (1−t) 1+α) , where α is a positive non-integer, are described. One algorithm is based on a knowledge of the Chebyshev series expansion of f on [−1, 1], the other two on polynomial interpolations to f at the zeros of TN, the Chebyshev polynomial of the first kind. Convergence theorems are given for each algorithm, and each algorithm is demonstrated numerically.

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