Difference between revisions of "Sokhotskii formulas"
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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 . | 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 | + | 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: |
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− | 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: | ||
− | + | <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> | |
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− | let | + | 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 |
− | be the interior of | ||
− | |||
− | 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> | |
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− | be a Cauchy-type integral. Then, for any point | + | 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 |
− | 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> | |
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− | + | <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> | |
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exist and are given by the Sokhotskii formulas | 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> | |
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or, equivalently, | 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> | |
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− | + | <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> | |
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− | The integrals along | + | 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. |
− | 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 | + | 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 |
− | 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 | ||
− | + | <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 | + | 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" />. |
− | the Sokhotskii formulas (2) and (3) remain valid for the interior points of the arc | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s086020a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s086020a.gif" /> | ||
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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 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 | + | 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" />. |
− | 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 [[#References|[6]]], [[#References|[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 [[#References|[7]]]. | For spatial analogues of the Sokhotskii formulas see [[#References|[7]]]. | ||
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====References==== | ====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> | <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==== | ====Comments==== | ||
In the Western literature one usually speaks of the Plemelj formulas; the combination Sokhotskii–Plemelj formulas also occurs. | 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 | + | 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]]]. |
− | 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 | + | 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]]). |
− | 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==== | ====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> | <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> |
Revision as of 14:53, 7 June 2020
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 |
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 [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 spaces. For further information see [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 can be represented as such jumps in an appropriate sense, cf. [a3]. The ultimate in this direction are Sato's hyperfunctions (cf. [a5] and Hyperfunction).
References
[a1] | A.P. Calderón, "Cauchy integrals on Lipschitz curves and related operators" Proc. Nat. Acad. Sci. USA , 74 : 4 (1977) pp. 1324–1327 |
[a2] | 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) |
[a3] | H.G. Tillmann, "Darstellung der Schwartzschen Distributionen durch analytischer Funktionen" Math. Z. , 77 (1961) pp. 106–124 |
[a4] | M. Sato, "Theory of hyperfunctions I-II" J. Fac. Sci. Univ. Tokyo I , 8 (1959–1960) pp. 139–193; 387–437 |
[a5] | L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) |
Sokhotskii formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sokhotskii_formulas&oldid=48744