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Hartogs triangle
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(Example)
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A non-trivial example of domain of holomorphy that has some interesting non-obvious properties is the Hartogs triangle which is the set
Since it is a Reinhardt domain it can be represented by plotting it on the plane
 as follows.
Figure 1: Hartogs triangle
It is obvious then where the name comes from. To see that this is a domain of holomorphy, then given a boundary point we wish to exhibit a holomorphic function on the whole Hartogs triangle which does not extend beyond that point. First note that on the top boundary  is anything and
 for some  , so
 will not extend beyond
 . Now for the diagonal boundary this is where
 , that is
 for some  , so
 will do not extend beyond
 .
One of the many properties of this domain is that if  is the Hartogs triangle, then it is a domain of holomorphy, but if we take a sufficently small neighbourhood  of  (the closure of  ), then any function holomorphic on  is holomorphic on the polydisc  (just fill in everything below the triangle in Figure 1). So if  does not include all of  then it is not a domain of holomorphy. This is because a Reinhardt domain that contains zero (the point  ) is a domain of holomorphy if and only if it is a logarithmically convex set and any neighbourhood of  does contain zero while  itself does not.
- 1
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"Hartogs triangle" is owned by jirka.
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Cross-references: logarithmically convex set, contains, triangle, polydisc, function, closure, neighbourhood, domain, diagonal, holomorphic function, point, boundary, obvious, plane, Reinhardt domain, properties, domain of holomorphy
There is 1 reference to this entry.
This is version 2 of Hartogs triangle, born on 2004-08-02, modified 2004-08-10.
Object id is 6059, canonical name is HartogsTriangle.
Accessed 1854 times total.
Classification:
| AMS MSC: | 32T05 (Several complex variables and analytic spaces :: Pseudoconvex domains :: Domains of holomorphy) |
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Pending Errata and Addenda
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![\includegraphics[scale=1.0]{hartogstriangle.eps}](http://images.planetmath.org:8080/cache/objects/6059/js/img3.png "\includegraphics[scale=1.0]{hartogstriangle.eps}")