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)

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.