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In this paper we consider the problem of computing 2-D Cauchy principal value integrals of the form  \[    ₯ S F ( P 0 ; P ) d P , P 0 ∈  S , {\fint _S}F({P_0};P)\,dP,\qquad {P_0} \in S,     \]  where  S  is either a rectangle or a triangle, and      F ( P 0 ; P ) F({P_0};P)       is integrable over  S , except at the point      P 0 {P_0}       where it has a second-order pole. Using polar coordinates, the integral is first reduced to the form  \[    ∫  θ  1 θ  2 [ u n k ] 0 R ( θ  ) f ( r , θ  ) r d r d θ  , \int _{{\theta _1}}^{{\theta _2}} {[unk]_0^{R(\theta )}\frac {{f(r,\theta )}}{r}dr\, d \theta ,}     \]  where      [ u n k ] [unk]       denotes the finite part of the (divergent) integral. Then ad hoc products of one-dimensional quadrature rules of Gaussian type are constructed, and corresponding convergence results derived. Some numerical tests are also presented.

The numerical evaluation of a 2-D Cauchy principal value integral arising in boundary integral equation methods