Computation of the roots of cauchy type principal value integrals

Abstract

A new method for the computation of the roots of Cauchy type principal value integrals (inside the integration interval) is proposed. This method is based on the extension of the formulae defining such integrals (by analytic continuation) outside the integration interval and, furthermore, on the application of the method of Burniston and Siewert for the computation of the zeros of sectionally analytic functions. A recent analogous method proposed by the author can alternatively be used. Gaussian quadrature rules are used for the numerical computation of the integrals and, hence, of the sought roots. Numerical results are presented for the roots of the Legendre function of the second kind Q₁(x).

Keywords:

• Analytic functions
• Cauchy type integrals
• Gaussian quadrature rules
• Legendre functions of the second kind
• principal value integrals
• Riemann‐Hilbert boundary value problem
• roots
• zeros

C.R. Categories::

• Primary, G.1.5
• secondary, G.1.4