Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives

Abstract

Algorithms are proposed for the numerical evaluation of Cauchy principal value integrals ⨍ − 1 1 w ( t ) f ( t ) / ( t − x ) d t , − 1 < x < 1 , with weight functions of Jacobi type singularities w ( t ) = ( 1 − t ) α ( 1 + t ) β , where α = ± 1 / 2 and β = ± 1 / 2 , for a given function f ( t ) and Hadamard finite-part integrals ⨎ − 1 1 w ( t ) f ( t ) / ( t − x ) 2 d t . The function f is interpolated by using a finite sum of Chebyshev polynomials. The present algorithms require O ( N log N ) arithmetic operations, where N is the order of the interpolation polynomial. It is shown that the present scheme gives uniform approximations, namely the errors are bounded independently of x , and is very efficient for smooth f . Further, we discuss approximations of hyper-singular integrals ∫ − 1 1 w ( t ) f ( t ) / ( t − x ) n d t , n ≥ 3 , and show their uniform convergences. Numerical examples are given to demonstrate the performance of the present schemes.