By means of the Riemann boundary value problem and of the properties of the Cauchy principal value integral we obtain the explicit expressions of general solution and their solvability conditions for these equations.
exists under the meaning of Cauchy principal value, and its value is equal to 1/2, where [partial derivative][OMEGA] and [theta](t) are the same as before.
Peng, "The first negative moment in the sense of the Cauchy principal value
," Statistics & Probability Letters, vol.
The definition of the potential related quantities in (15) and (16) capturing the Cauchy Principal value of the surface integrals are
Furthermore, [[[nabla][PHI]].sup.n.sub.CPV], the other potential magnitude in (27), denotes the contribution of the n-th div-TO basis function to the Cauchy Principal value of the gradient of the electric scalar potential.
Sine methods, Hilbert transform, Cauchy principal value
The a's and b's may be complex (with the b's not equal to zero), but the integral is assumed to be well defined, possibly as a Cauchy principal value. In particular the line segment with endpoints [a.sub.i], + [b.sub.i]x and [a.sub.i] + [b.sub.i]y is assumed to lie in the cut plane C\(-[infinity], 0) for 1 [less than or equal to] [less than or equal to] h.
If the last variable of [R.sub.J] is negative, the Cauchy principal value is given by
Calculations were performed with the package MATHEMATICA, matrices D(x, y) and P(x, y) were computed symbolically, the approximate solutions of the boundary integral equations were obtained by means of cubic splines, and the Cauchy principal values
were computed with Gaussian quadrature.