Hartogs-Laurent series

A series

$$ \tag{* } \sum _ {k = - \infty } ^ \infty f _ {k} ( \prime z) ( z _ {n} - a _ {n} ) ^ {k} , $$

where $ \prime z = ( z _ {1} \dots z _ {n-} 1 ) $ and where the $ f _ {k} ( \prime z) $ are functions holomorphic in some domain $ \prime D \subset \mathbf C ^ {n-} 1 $ which is independent of $ k $. If $ f _ {k} = 0 $ for all $ k < 0 $, the series (*) is known as a Hartogs series. Any holomorphic function in a Hartogs domain $ D $ of the type

$$ \{ {( \prime z, z _ {n} ) } : { \prime z \in \prime D,\ 0 \leq r ( \prime z) < | z _ {n} - a _ {n} | < R ( \prime z) \leq + \infty } \} $$

can be expanded into a Hartogs–Laurent series which converges absolutely and uniformly inside $ D $. In complete Hartogs domains this will be the expansion into a Hartogs series. The domains of convergence of Hartogs–Laurent series are domains of the same kind with special $ r ( \prime z) $ and $ R ( \prime z) $, known as Hartogs radii. If $ n = 1 $, when all $ f _ {k} $ are constant, a Hartogs–Laurent series is called a Laurent series.

References

[1]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

References

[a1]	H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) (Elraged & Revised Edition. Original: 1934)
[a2]	S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948)