# Schwarz kernel

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 for the 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

 [1] 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) [2] S.G. Mikhlin, "Linear integral equations" , Hindushtan Publ. Comp. , Delhi (1960) (Translated from Russian)

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.