View source for Bochner-Martinelli representation formula

''Bochner–Martinelli representation, Bochner–Martinelli formula''

An integral representation for holomorphic functions, which is defined as follows [[#References|[1]]], [[#References|[2]]]. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167201.png" /> be holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167202.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167203.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167204.png" /> be continuous in its closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167205.png" />. Then the expression

<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/b/b016/b016720/b0167206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</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/b/b016/b016720/b0167207.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/b/b016/b016720/b0167208.png" /></td> </tr></table>

here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167209.png" /> means that the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672010.png" /> is to be omitted, is called the Bochner–Martinelli representation. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672011.png" /> this representation is identical with the Cauchy integral formula (cf. [[Cauchy integral|Cauchy integral]]), but for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672012.png" /> its kernel is not holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672013.png" />, which is the reason for the limited applicability of the Bochner–Martinelli representation in the theory of functions of several complex variables. The kernel of the Bochner–Martinelli representation is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672014.png" />-differential form of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672015.png" />:

<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/b/b016/b016720/b01672016.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/b/b016/b016720/b01672017.png" /></td> </tr></table>

which is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672018.png" />, has a singular point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672019.png" />, and is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672021.png" />-closed (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672022.png" />) outside the singular point. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672023.png" />, the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672024.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672025.png" />, where

<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/b/b016/b016720/b01672026.png" /></td> </tr></table>

is a form of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672027.png" />, the coefficient of which is a fundamental solution of the Laplace equation; here

<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/b/b016/b016720/b01672028.png" /></td> </tr></table>

The following integral representation, which generalizes formula (1), is the analogue of the Cauchy–Green formula (cf. [[Cauchy integral|Cauchy integral]]): If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672029.png" /> is continuously differentiable in the closure of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672030.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672031.png" />, then, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672032.png" />,

<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/b/b016/b016720/b01672033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

The function

<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/b/b016/b016720/b01672034.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672035.png" /> is a smooth hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672037.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672038.png" /> which is Lebesgue-integrable, is said to be an integral of Bochner–Martinelli type. As for Cauchy-type integrals, Sokhotskii's formula, with the usual restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672040.png" />, is applicable to Bochner–Martinelli-type integrals. A Bochner–Martinelli-type integral is a complex function which is harmonic everywhere outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672041.png" />; in the general case this function is holomorphic only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672042.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672043.png" />, then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672044.png" />, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672045.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672046.png" /> is equivalent to the holomorphy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672047.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672048.png" />.

The Bochner–Martinelli representation is employed to demonstrate other integral representations (e.g. the [[Bergman–Weil representation|Bergman–Weil representation]]), in holomorphic continuation from the boundary, and also in the theory of boundary values of holomorphic functions of several complex variables. It was introduced by S. Bochner [[#References|[1]]] and by E. Martinelli [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Bochner,   "Analytic and meromorphic continuation by means of Green's formula"  ''Ann. of Math. (2)'' , '''44''' :  4  (1943)  pp. 652–673</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Martinelli,   ''Rend. Accad. Ital.'' , '''9'''  (1938)  pp. 269–283</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>



====Comments====
Sokhotskii's formula is also known as Plemelj's formula. The Cauchy–Green formula is also known as the Cauchy–Pompeiu formula.

The Bochner–Martinelli kernel is a special case of a Cauchy–Fantappié kernel. The integral representation (2) solves the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672050.png" />-equation:

<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/b/b016/b016720/b01672051.png" /></td> </tr></table>

for compactly-supported <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672052.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672053.png" />, by substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672054.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672055.png" /> in the right-hand side of (2) and omitting the integral over the boundary. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672056.png" /> is not compactly supported, the boundary integral causes difficulties. These can be solved for strictly pseudo-convex domains and the Bochner–Martinelli kernel then occurs in an explicit solution operator for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672057.png" />-equation.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,   "Function theory of several complex variables" , Wiley  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. [G.M. Khenkin] Henkin,   J. Leiterer,   "Theory of functions on complex manifolds" , Birkhäuser  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.M. Range,   "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. 1, Sect. 3</TD></TR></table>