By Marek Jarnicki and Peter Pflug

This ebook presents a entire advent to the sector of numerous advanced variables within the atmosphere of a truly certain yet easy category of domain names, the so-called Reinhardt domain names. during this manner the reader may possibly research a lot approximately this quarter with out encountering too many technical problems. bankruptcy 1 describes the elemental notions and the phenomenon of simultaneous holomorphic extension. bankruptcy 2 offers a pretty entire dialogue of biholomorphisms of bounded (complete) Reinhardt domain names within the dimensional case. The 3rd bankruptcy provides a type of Reinhardt domain names of life for an important sessions of holomorphic features. The final bankruptcy bargains with invariant services and offers specific calculations of lots of them on definite Reinhardt domain names. quite a few workouts are integrated to assist the readers with their figuring out of the cloth. additional effects and open difficulties are further that may be worthwhile as seminar themes. the first goal of this publication is to introduce scholars or non-experts to a few of the most examine components in different complicated variables. The ebook presents a pleasant invitation to this box because the basically prerequisite is a uncomplicated wisdom of study.

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Additional resources for First Steps in Several Complex Variables: Reinhardt Domains

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F . f . 16 (Cauchy inequalities). a;r/ ; ˛ 2 ZnC : r˛ Similarly as in the case of one complex variable, the following results are easy consequences of the Cauchy inequalities (Exercise). 17 (Liouville theorem). Cn /, k 2 ZC . z/j Ä C kzkk for kzk R0 . 18. a; r/. 7. 19 (Weierstrass theorem). ˝/ 3 f ! ˝/ and D ˛f ! D ˛f locally uniformly on ˝ for any ˛ 2 ZnC . 20. Let D Then f is an open mapping. D/, f 6Á const. Proof. Fix an a 2 D. By the identity principle, there exists an X 2 Cn such that the function g S 3 7!

WD † Rn . ˛/ 3 z 7! ˛/ where jzn j˛n 2 RnC is well defined (here 00 WD 1). Notice that in the case where ˛ 2 Zn , jz ˛ j coincides with the absolute value of z ˛ . 5. Reinhardt domains Observe that D˛;c is a Reinhardt domain (Exercise). It is called an elementary Reinhardt domain. We put D˛ WD D˛;0 . ˛; c/ for some > 0. 3. 1;1/ and D. jz2 j 1; 1/ . 4. 2; jz1 j 1/ and D. 2;1/ . 7. Rn / . j / D D˛;c ” D˛;c \ Vj ¤ ¿ ” ˛j 0: ( 18 Note that D0;c D Cn ¿; if c > 0; if c Ä 0: 34 Chapter 1. 5. Elementary domains D˛;c with (˛1 D 0; ˛2 > 0) and (˛1 D 0; ˛2 < 0).

7. a0 ; a00 / 2 Ck Cn k ). a; r//. Proof. It is obvious that g is continuous. z 0 ; / @zj e C1 . z; / 7! z; / @zj 50 Chapter 1. Reinhardt domains is continuous, j D 1; : : : ; n. a; r/; j D 1; : : : ; n; exist. 8. a00 ; r 00 / ! a; r/. 9. (a) Holomorphic functions are infinitely differentiable in the complex sense. ˝/ for arbitrary ˛ 2 ZnC . 10 (Identity principle). D/, where D domain. z/g ¤ ¿. Proof. Clearly (i) ) (ii) , (iii). Since D is connected, to prove the implication (ii) ) (i) it is sufficient to note that the set D0 WD fz 2 D W Tz f D Tz gg is non-empty open and closed in D.

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