The numerical evaluation of Cauchy principal value integrals via the fast Fourier transform

Abstract

The numerical evaluation of Cauchy principal value integrals is largely based on quadrature rules of primitive or Gauss-type which provide accurate solutions for selected points of the abscissae. An alternative method is proposed whereby the Cauchy principal value integral is expressed first as the Hilbert transform of the density functionf(t), then as an expression which involves sine and cosine transforms of f(t) which can be evaluated with the fast Fourier transform (FFT). An example is considered which compares the FFT method with Ivanov's and Laguerre quadrature methods, and it is shown that although the quadrature methods obtain accurate solutions, they are relatively inefficient if solutions are sought at numerous locations along the abscissae.

Keywords:

• Singular integral equations
• Hilbert transforms
• fast Fourier transforms

C.R. Categories:

• Primary: G.1.4
• Secondary: F.2.1

†This research has been suppoted by a grant from the American Oil Company, Tulsa, Oklahoma.

†This research has been suppoted by a grant from the American Oil Company, Tulsa, Oklahoma.

Notes

†This research has been suppoted by a grant from the American Oil Company, Tulsa, Oklahoma.