By Marek Jarnicki

This monograph is dedicated to a scientific exposition of the idea of extension of holomorphic features, e. g. characterizations of envelopes of holomorphy with admire to a variety of households of holomorphic capabilities. as a result, there's emphasis on an in depth presentation of holomorphic convexity and pseudoconvexity of Riemann domain names over Cn.

Our curiosity during this sector of complicated research all started without delay after our reviews while either one of us have been attracted to continuation of holomorphic features. through the years we received the influence that there's a have to have a resource the place the most effects might be came across. we are hoping this publication can function this type of resource. the alternative of issues evidently displays our own personal tastes.

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**Example text**

The mapping J* TatE the ring of all powerseries with centerata that are convergent in a neighborhood of a is an isomorphism. Put (9:= U x {a} a cC" a)) = a (in the sequel we wiLl denote elements of (9 either formally as pairs (f, a) (when we want to point out that or simply as germs f). For and for any (U, f) f define and let ir: (9 —+ C" be given by the formula t11(U,f) z E U} C (9. f)Ef is a neighborhood basis 16)• We endow (9 with the topology generated by this basis. We will show that this is a Hausdorif topology.

Take an arbitrary neighborhood V E C V. Then U C e(a. p is injective. Suppose that = 3(a'). In particular, of connected neighborhoods of yo with = =: yo. Fix a basis k > I, and let xk E Uk := e(a, Vk), E (4 := 1, Vk), k Vk+I C We know that; := := '. 4 can be connected in çH(V). Let k > k0 be such that Vk C V. Then Uk U (4 C U. Hence e(a', V) = U and therefore U E a'. Thus a = a'. It remains to show that is surjective. Let; = E 6. (xk) there exists exactly one connected component, say of such that Xk E fork >> 1.

5 The boundary of a Riemann domain First we prove a' a" and that there are a'. a" E U such that =: = yo. 7(a) there exist U' E may assume that a', a" E U fl We 37 with U' fl U" = 0. By (*) we may assume that U' = e(a', V'), U" = e(a", V') with (I' U U" c U0 and V' c V. Observe that V' \ P is connected and U" E a" therefore 0 \ P) C U' fl U"; contradiction. is continuous. Take a E U fl dX and let yo := Now we prove that 0. Hence V) and let U1 := e(a. V,). By (*) U1 fl Uo V1 Wehavetoshow \P) CU1.