Three algorithms for Hadamard finite-part integrals and fractional derivatives

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.