View source for Schwarz kernel

<!--
s0835501.png
$#A+1 = 30 n = 0
$#C+1 = 30 : ~/encyclopedia/old_files/data/S083/S.0803550 Schwarz kernel
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''in the disc  $  | z | < 1 $''

The function

$$ 
T( z;  \zeta )  =  
\frac{\zeta + z }{\zeta - z }
 ,\ \ 
\zeta = e ^ {i \sigma } ,\ \ 
0 \leq  \sigma \leq  2 \pi .
$$

Let  $  D $
be a finite simply-connected or multiply-connected domain with boundary  $  \Gamma $, 
let  $  G( z;  \zeta ) $
be the [[Green function|Green function]] for the [[Laplace operator|Laplace operator]] in  $  D $, 
and let the real-valued function  $  H( z;  \zeta ) $
be the conjugate to  $  G( z;  \zeta ) $. 
Then the function  $  M( z;  \zeta ) = G( z;  \zeta ) + iH( z;  \zeta ) $
is called the complex Green function of the domain  $  D $. 
The function  $  M( z;  \zeta ) $
is an analytic but multiple-valued (if  $  D $
is multiply connected) function of  $  z $
and a single-valued non-analytic function of  $  \zeta $. 
The function

$$ 
T( z;  \zeta )  =  
\frac{\partial  M( z;  \zeta ) }{\partial  \nu }
 ,
$$

where  $  \nu $
is the direction of the interior normal at  $  \zeta \in \Gamma $, 
is called the Schwarz kernel of  $  D $.

Let  $  F( z) = u( z) + iv( z) $
be an analytic function without singular points in  $  D $, 
and let  $  u $
be single valued and continuous in  $  D \cup \Gamma $. 
Then the following formula holds:

$$ 
F( z)  =  
\frac{1}{2 \pi }
 \int\limits _  \Gamma  u( \zeta ) T( z;  \zeta )  d \sigma + iv( a),
$$

where  $  a \in D $
is a fixed point and  $  v( a) $
is the value at  $  a $
of one of the branches of the function  $  v( z) $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.P. Zabreiko (ed.)  A.I. Koshelev (ed.)  M.A. Krasnoselskii (ed.)  S.G. Mikhlin (ed.)  L.S. Rakovshchik (ed.)  V.Ya. Stet'senko (ed.)  T.O. Shaposhnikova (ed.)  R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Mikhlin,   "Linear integral equations" , Hindushtan Publ. Comp. , Delhi  (1960)  (Translated from Russian)</TD></TR></table>

====Comments====
Of course, some regularity conditions on  $  \Gamma $
have to be assumed, so that the normal derivative  $  ( \partial  M )/ ( \partial  \nu ) $
is well defined. Note that the real part of  $  T $
is the Poisson kernel.

See also [[Schwarz integral|Schwarz integral]].