View source for Sokhotskii formulas

Formulas first discovered by Yu.V. Sokhotskii [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860201.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860204.png" />, be a closed smooth Jordan curve in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860205.png" />-plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860206.png" /> be the complex density in a Cauchy-type integral along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860207.png" />, on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860208.png" /> satisfies a Hölder condition:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860209.png" /></td> </tr></table>

let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602010.png" /> be the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602012.png" /> its exterior, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

be a Cauchy-type integral. Then, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602014.png" /> the limits

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602015.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602016.png" /></td> </tr></table>

exist and are given by the Sokhotskii formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

or, equivalently,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602018.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602019.png" /></td> </tr></table>

The integrals along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602020.png" /> 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602021.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602022.png" />) as values of the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602024.png" />, one thus obtains a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602025.png" /> that is continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602026.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602027.png" />). In the large, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602028.png" /> is sometimes described as a piecewise-analytic function.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602031.png" /> are also Hölder continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602032.png" /> with the same exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602033.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602034.png" />, with any exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602035.png" />. For the corner points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602036.png" /> of a piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602037.png" /> (see Fig.), the Sokhotskii formulas take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

In the case of a non-closed piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602039.png" />, the Sokhotskii formulas (2) and (3) remain valid for the interior points of the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602040.png" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s086020a.gif" />

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 [[#References|[3]]], [[#References|[5]]]), and also in solving various applied problems in function theory (see [[#References|[4]]]).

The question naturally arises as to the possibility of extending the conditions on the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602041.png" /> and the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602042.png" />, 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 [[#References|[6]]], [[#References|[8]]]). For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602043.png" /> be a rectifiable Jordan curve, and let the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602044.png" /> be, as before, Hölder continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602045.png" />. Then the Sokhotskii formulas (2) hold almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602048.png" /> are understood as non-tangential boundary values of the Cauchy-type integral from inside and outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602049.png" />, respectively, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602051.png" /> are, in general, not continuous in the closed domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602053.png" />.

For spatial analogues of the Sokhotskii formulas see [[#References|[7]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Sokhotskii,   "On definite integrals and functions used in series expansions" , St. Petersburg  (1873)  (In Russian)  (Dissertation)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  J. Plemelj,   "Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend"  ''Monatsh. Math. Phys.'' , '''19'''  (1908)  pp. 205–210</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  J. Plemelj,   "Riemannsche Funktionenscharen mit gegebener Monodromiegruppe"  ''Monatsh. Math. Phys.'' , '''19'''  (1908)  pp. 211–245</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. Muskhelishvili,   "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Muskhelishvili,   "Some basic problems of the mathematical theory of elasticity" , Noordhoff  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F.D. Gakhov,   "Boundary value problems" , Pergamon  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.V. Bitsadze,   "Fundamentals of the theory of analytic functions of a complex variable" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
In the Western literature one usually speaks of the Plemelj formulas; the combination Sokhotskii–Plemelj formulas also occurs.

Much work has been done on Cauchy-type integrals in recent years. Thus, A.P. Calderón [[#References|[a1]]] has proved the existence of non-tangential boundary values almost everywhere under weaker conditions, and he has studied the principal value integral, or Cauchy transform, in (2) as a bounded linear operator on appropriate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602054.png" /> spaces. For further information see [[#References|[a2]]].

The idea of representing a function on a simple closed curve as the jump of an analytic function across that curve has been extensively generalized. Thus, every distribution on the unit circle and every tempered distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602055.png" /> can be represented as such jumps in an appropriate sense, cf. [[#References|[a3]]]. The ultimate in this direction are Sato's hyperfunctions (cf. [[#References|[a5]]] and [[Hyperfunction|Hyperfunction]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.P. Calderón,   "Cauchy integrals on Lipschitz curves and related operators"  ''Proc. Nat. Acad. Sci. USA'' , '''74''' :  4  (1977)  pp. 1324–1327</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.P. Khavin (ed.)  S.V. Khrushchev (ed.)  N.K. Nikol'skii (ed.) , ''Linear and complex analysis problem book'' , ''Lect. notes in math.'' , '''1043''' , Springer  (1984)  pp. Chapt. 6  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.G. Tillmann,   "Darstellung der Schwartzschen Distributionen durch analytischer Funktionen"  ''Math. Z.'' , '''77'''  (1961)  pp. 106–124</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Sato,   "Theory of hyperfunctions I-II"  ''J. Fac. Sci. Univ. Tokyo I'' , '''8'''  (1959–1960)  pp. 139–193; 387–437</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>