Hartogs triangle
A non-trivial example of domain of holomorphy that has some interesting non-obvious properties is the Hartogs triangle which is the set
{(z,w)∈ℂ2∣|z|<|w|<1}. |
Since it is a Reinhardt domain it can be represented by plotting it on the plane |z|×|w| 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 z is anything and w=eiθ for some θ, so
f(z,w)=1w-eiθ will not extend beyond (z,eiθ).
Now for the diagonal boundary this is where |z|=|w|,
that is z=eiθw for some θ, so
f(z,w)=1z-eiθw will do not extend beyond (eiθw,w).
One of the many properties of this domain is that if U is the Hartogs triangle, then it is a domain of holomorphy, but if we take a sufficently small neighbourhood V of ˉU (the closure of U), then any function holomorphic on V is holomorphic on the polydisc D2(0,1) (just fill in everything below the triangle in Figure 1). So if V does not include all of D2(0,1) then it is not a domain of holomorphy. This is because a Reinhardt domain that contains zero (the point (0,0)) is a domain of holomorphy if and only if it is a logarithmically convex set and any neighbourhood of ˉU does contain zero while U itself does not.
References
- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title | Hartogs triangle |
---|---|
Canonical name | HartogsTriangle |
Date of creation | 2013-03-22 14:31:08 |
Last modified on | 2013-03-22 14:31:08 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 5 |
Author | jirka (4157) |
Entry type | Example |
Classification | msc 32T05 |