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An automatic quadrature is presented for computing Cauchy principal value integrals Q(f; c) = fa f(t)/(t-c)dt, a < c < b, for smooth functions f(t). After subtracting out the singularity, we approximate the function f(t) by a sum of Chebyshev polynomials whose coefficients are computed using the FFT. The evaluations of Q(f; c) for a set of values of c in (a, b) are efficiently accomplished with the same number of function evaluations. Numerical examples are also given. k=0

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An automatic quadrature method is presented for approximating fractional derivative D q f (x) of a given function f (x), which is defined by an indefinite integral involving f (x). The present method interpolates f (x) in terms of the Chebyshev polynomials in the range [0, 1] to approximate the fractional derivative D q f (x) uniformly for 0 ≤ x ≤ 1, namely the error is bounded independently of x. Some numerical examples demonstrate the performance of the present automatic method.

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