Sokhotskii formulas

Formulas first discovered by Yu.V. Sokhotskii , expressing the boundary values of a Cauchy-type integral. With more complete proofs but significantly later, the formulas were obtained independently by J. Plemelj .

Let : , , , be a closed smooth Jordan curve in the complex -plane, let be the complex density in a Cauchy-type integral along , on which satisfies a Hölder condition: let be the interior of , its exterior, and let (1)

be a Cauchy-type integral. Then, for any point the limits  exist and are given by the Sokhotskii formulas (2)

or, equivalently,  The integrals along on the right-hand sides of these formulas are understood in the sense of the Cauchy principal value and are so-called singular integrals. By taking, under these conditions, (or ) as values of the integral on , one thus obtains a function that is continuous in the closed domain (respectively, ). In the large, is sometimes described as a piecewise-analytic function.

If , then and are also Hölder continuous on with the same exponent , while if , with any exponent . For the corner points of a piecewise-smooth curve (see Fig.), the Sokhotskii formulas take the form (3)

In the case of a non-closed piecewise-smooth curve , the Sokhotskii formulas (2) and (3) remain valid for the interior points of the arc . Figure: s086020a

The Sokhotskii formulas play a basic role in solving boundary value problems of function theory and in the theory of singular integral equations (see , ), and also in solving various applied problems in function theory (see ).

The question naturally arises as to the possibility of extending the conditions on the contour and the density , in such a way that the Sokhotskii formulas (possibly with some restrictions) remain valid. The most significant results in this direction are due to V.V. Golubev and I.I. Privalov (see , ). For example, let be a rectifiable Jordan curve, and let the density be, as before, Hölder continuous on . Then the Sokhotskii formulas (2) hold almost everywhere on , where and are understood as non-tangential boundary values of the Cauchy-type integral from inside and outside , respectively, but and are, in general, not continuous in the closed domains and .

For spatial analogues of the Sokhotskii formulas see .