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

An integral representation for holomorphic functions, which is defined as follows [[#References|[1]]], [[#References|[2]]]. Let the function  $  f $
be holomorphic in a domain  $  D \subset  {\mathbf C  ^ {n} } $
with piecewise-smooth boundary  $  \partial  D $, 
and let  $  f $
be continuous in its closure  $  \overline{D}\; $. 
Then the expression

$$ \tag{1 }

\frac{(n-1)! }{(2 \pi i)  ^ {n} }

\int\limits _ {\partial  D }

\frac{f( \zeta ) }{| \zeta -z |  ^ {2n} }
 \times
$$

$$ 
\times 
\sum _ { j=1 } ^ { n }  ( \overline \zeta \; _ {j} - \overline{z}\; _ {j} ) \ 
d \overline \zeta \; _ {1}  \wedge  d \zeta _ {1}  \wedge \dots \wedge \ 
[d \overline \zeta \; _ {j} ]  \wedge  d \zeta _ {j}  \wedge \dots
\wedge  d \zeta _ {n\ } =
$$

$$ 
= \ 
\left \{ 
\begin{array}{cl}
f(z),  & \textrm{ if }
 z \in D,  \\
0,  & \textrm{ if }  z \notin \overline{D}\; ,  \\
\end{array}

 \right .$$

here  $  [d \overline \zeta \; _ {j} ] $
means that the term  $  d \overline \zeta \; _ {j} $
is to be omitted, is called the Bochner–Martinelli representation. For  $  n = 1 $
this representation is identical with the Cauchy integral formula (cf. [[Cauchy integral|Cauchy integral]]), but for  $  n > 1 $
its kernel is not holomorphic in  $  z $, 
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  $  \zeta $-
differential form of bidegree  $  (n, n - 1) $:

$$ 
\omega ( \zeta , z )  = \ 
(n-1)! over {(2 \pi i)  ^ {n} }

\frac{1}{| \zeta -z |  ^ {2n} }
 \times
$$

$$ 
\times 
\sum _ { j=1 } ^ { n }  ( \overline \zeta \; _ {j} - \overline{z}\; _ {j} )
 d \overline \zeta \; _ {1}  \wedge  d \zeta _ {1}  \wedge \dots
\wedge  [ d \overline \zeta \; _ {j} ]  \wedge
 d \zeta _ {j}  \wedge \dots \wedge  d \overline \zeta \; _ {n}  \wedge  d \zeta _ {n} ,
$$

which is defined in  $  \mathbf C  ^ {n} $, 
has a singular point at  $  \zeta = z $, 
and is  $  \overline \partial \; $-
closed (i.e.  $  \overline \partial \; \omega = 0 $) 
outside the singular point. If  $  n > 1 $, 
the form  $  \omega $
is equal to  $  \partial  \omega  ^  \prime  ( \zeta , z) $, 
where

$$ 
\omega  ^  \prime  ( \zeta , z)  =  -
(n-2)! over {(2 \pi i)  ^ {n} } \cdot

\frac{1}{| \zeta -z |  ^ {2n-2} }

\sum _ { j=1 } ^ { n } 
\left ( \prod _ { k\neqj }
d \overline \zeta \; _ {k}  \wedge  d \zeta _ {k} \right )
$$

is a form of bidegree  $  (n - 1, n - 1) $, 
the coefficient of which is a fundamental solution of the Laplace equation; here

$$ 
\partial  \phi  = \ 
\sum dz _ {k} 
\frac{\partial  \phi }{\partial  z _ {k} }
 \ 
 \textrm{ and } \ \ 
\overline \partial \; \phi  = \ 
\sum d \overline{z}\; _ {k} 
\frac{\partial  \phi }{\partial  z _ {k} }
 .
$$

The following integral representation, which generalizes formula (1), is the analogue of the Cauchy–Green formula (cf. [[Cauchy integral|Cauchy integral]]): If the function  $  f $
is continuously differentiable in the closure of a domain  $  D \subset  \mathbf C  ^ {n} $
with piecewise-smooth boundary  $  \partial  D $, 
then, for any point  $  z \in D $,

$$ \tag{2 }
f(z)  =  \int\limits _ {\partial  D }
f( \zeta ) \omega ( \zeta , z) -
\int\limits _ { D } \overline \partial \; f( \zeta )
\wedge \omega ( \zeta , z).
$$

The function

$$ 
\widehat{f}  (z)  = \ 
\int\limits _  \Gamma  f( \zeta )
\omega ( \zeta , z),
$$

where  $  \Gamma $
is a smooth hypersurface in  $  \mathbf R  ^ {2n} = \mathbf C  ^ {n} $
and  $  f $
is a function on  $  \Gamma $
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  $  \Gamma $
and  $  f $, 
is applicable to Bochner–Martinelli-type integrals. A Bochner–Martinelli-type integral is a complex function which is harmonic everywhere outside  $  \Gamma $; 
in the general case this function is holomorphic only for  $  n = 1 $. 
If  $  \Gamma = \partial  D $, 
then if  $  n \geq  1 $, 
the condition  $  \widehat{f}  (z) \equiv 0 $
outside  $  \overline{D}\; $
is equivalent to the holomorphy of  $  \widehat{f}  $
in  $  D $.

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  $  \overline \partial \; $-
equation:

$$ 
\overline \partial \; f  = \ 
v \  \textrm{ with } \ \ 
\overline \partial \; v  =  0,
$$

for compactly-supported  $  (0, 1) $-
forms  $  v $, 
by substituting  $  v $
for  $  \partial  f $
in the right-hand side of (2) and omitting the integral over the boundary. When  $  v $
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  $  \overline \partial \; $-
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>