Regularization of the Divergent Integrals. II. Application in Fracture Mechanics.

In this article the methodology for divergent integral regularization developed in [8] is applied for regularization of the weakly singular and hypersingular integrals, which arise when the boundary integral equations (BIE) methods are used to solve problems in fracture mechanics. The approach is based on the theory of distribution and the application of the Green theorem. The weakly singular and hypersingular integrals over arbitrary convex polygon have been transformed to the regular contour integrals that can be easily calculated analytically or numerically.

Quadrature relations for finite-limit, principal-value integrals

An intuitive expression is obtained for a general finite-limit, prinicipal-value (PV) integral containing a pole of arbitrary order. Such an integral is simply the difference between its "end-point quadratures," the quadrature being the result of the indefinite integral. This PV expression is fairly obvious for simple poles but requires careful justification for higher order poles. Moreover, quadratures are useful in evaluating related integrals that contain both poles and other singularities (e.g., step functions or logarithmic divergences). Then, for a certain type of finite-limit divergent integral, the PV is interpreted as its "convergent part." Also, for cases in which there are two PV's in a double integral, it is shown that the famous Poincaré–Bertrand theorem applies to finite-limit as well as infinite-limit integrals. Finally, an interesting quadrature relation is derived for such double integrals, and the validity of the Poincaré–Bertrand theorem is explicitly demonstrated for a simple finite-limit case.

Evaluation of a class of integrals occurring in mathematical physics via a higher order generalization of the principal value

The notion of the principal value of an integral is generalized to treat higher order singularities. The principal value of an integral can be considered the "convergent part" of a divergent integral, an interpretation that is almost trivial for simple poles, but more meaningful for higher order poles. Application of this concept leads to a simple algorithm that may be applied to the evaluation of a class of integrals arising in mathematical physics. Many of these integrals frequently occur in the analytic and numerical evaluation of folding functions arising from the product of single-particle Green's functions.

The anomalous moments of nucleons

The formalism of the preceding paper is applied to calculate the magnetic moments of nucleons and the neutron-electron interaction in the symmetric PS-PS theory. Vacuum polarization is ignored, and a very simple trial function employed. With a suitable coupling constant, and a reasonable cut-off for the single logarithmically divergent integral that appears, a rather rough fit to the observed values is obtained.