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By So-chin Chen


Abstract. This is an expository article concerning complex analysis, in particular, several complex variables. Several subjects are discussed here to demonstrate the development and the diversity of several complex variables. Hopefully, the brief introduction to complex analysis in several variables would motivate the reader’s interests to this subject. The purpose of this article is to give a brief expository introduction to complex analysis, in particular, to several complex variables. Complex analysis differs dramatically between one and several variables. Many fundamental features change when space dimension jumps from one to greater than one. For instance, any domain D on the complex plane is a domain of holomorphy, that is, there is a holomorphic function f on D which cannot be extended holomorphically across any boundary point of D. In Cn, n ≥ 2, this is not the case. It is easy to construct a domain D ⊂ Cn, n ≥ 2, such that any holomorphic function f on D extends holomorphically to a fixed strictly larger domain D1 containing D. Another effect is that the set of singularities of a meromorphic function g in C is always discrete, e.g., g(z) = 1/z has a pole at zero and is holomorphic otherwise. Such an effect cannot happen in several variables due to the Hartogs extension theorem. As a matter of fact, every function f holomorphic on D \ K, where K is a compact subset of D such that D \ K is connected, extends holomorphically to D. Also, there does not exist an analog of the famous Riemann mapping theorem for one variable in higher-dimensional spaces. Even the open unit ball and polydisc in Cn, n ≥ 2, are not biholomorphically equivalent. This fundamental discovery is due to H

Year: 2000
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