Numerical evaluation of a Cauchy principal value integral that arises in a problem involving the generation of instability waves
The purpose of the present paper is to compute the principal value of an integral whose integrand is given in terms of Cauchy principal value integrals. For the integrand the author applies analytic methods as far as possible, using some special functions such as Dilogarithms. The main integral has the form \(\oint^{\infty}_{0}\Phi (t)/(t-1)dt\), with \(\Phi\) (1)\(\neq 0\). He first analyzes the behaviors of \(\Phi\) (t) at 0 and \(+\infty\), and then divides the interval \([0,+\infty)\) into three parts, say [0,1/2],[1/2,3/2], and \([3/2,+\infty)\). For the second interval involving principal value, he applies the Longman's idea \[ I=\oint^{a}_{-a}F(x)/x dx=\oint^{a}_{-a}[F(x)-F(-x)]/2x dx \] and uses Gauss-Legendre integration formula with even number of nodes to avoid the functional value 0/0 at the origin (corresponding to \(x=1)\). The result seems quite satisfactory.
scientific article; zbMATH DE number 3972060
scientific article