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


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 functionMathworldPlanetmath 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 U¯ (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 U¯ does contain zero while U itself does not.


  • 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