# Sokhotskii formulas

Formulas first discovered by Yu.V. Sokhotskii [1], 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 [3], [5]), and also in solving various applied problems in function theory (see [4]).

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 [6], [8]). 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 [7].

#### References

 [1] Yu.V. Sokhotskii, "On definite integrals and functions used in series expansions" , St. Petersburg (1873) (In Russian) (Dissertation) [2a] J. Plemelj, "Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend" Monatsh. Math. Phys. , 19 (1908) pp. 205–210 [2b] J. Plemelj, "Riemannsche Funktionenscharen mit gegebener Monodromiegruppe" Monatsh. Math. Phys. , 19 (1908) pp. 211–245 [3] N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian) [4] N.I. Muskhelishvili, "Some basic problems of the mathematical theory of elasticity" , Noordhoff (1975) (Translated from Russian) [5] F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian) [6] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) [7] A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian) [8] B.V. Khvedelidze, "The method of Cauchy-type integrals in the discontinuous boundary value problems of the theory of holomorphic functions of a complex variable" J. Soviet Math. , 7 (1977) pp. 309–415 Itogi Nauk. Sovremen. Probl. Mat. , 7 (1975) pp. 5–162