Reconciling two definitions of domains of holomorphy I have seen the following two definitions of a domain of holomorphy;  I was wondering if they are actually equivalent:
1) A domain $\Omega$ is a domain of holomorphy if there exists a holomorphic function on $\Omega$ that does not extend to a larger domain $\Omega' \supset \Omega$;
2) A domain $\Omega$ is a domain of holomorphy if for every domain $\Omega'$ which intersects the boundary of $\Omega$, and for every connected component $U$ of $\Omega \cap \Omega'$, there exists a holomorphic function $\varphi$ on $\Omega$ such that $\varphi|_U$ does not extend to $\Omega'$.
I think that at least the second condition implies the first; but does the first imply the second?
 A: The first is not a good definition.  Define
$\Omega \subset {\mathbb C}^2$ in the following way:
$$
\Omega = \{ (z,w) \mid z e^{i|w|^2} \text{is not a nonnegative real number} \} 
$$
For each fixed $w$, the set is simply a branch cut of the plane.  It is not difficult to show that for the complex line given by $z = -r$ for a negative real number $-r$ hits the boundary of $\Omega$ at a single point.  That is the boundary of omega except for the set $z=0$ is actually a strictly pseudoconvex hypersurface.   But it is $\Omega$ on both sides of it.
Take any $f$ holomorphic on $\Omega$.  You can see that according to the second (correct) definition, $\Omega$ is not a domain of holomorphy.  The $f$ will extend from the pseudoconcave side.  You should also note how the "every connected component" plays into this definition.
On the other hand, there clearly is not a large domain $\Omega'$ for the first definition.  You can just take a branch of $\log z$ so that it is holomorphic in $\Omega$.  That will be discontinuous at the branch cuts and so cannot extend.
The issue is connected to the idea of there not being a "largest domain of holomorphy containing a domain", just like there isn't a largest domain onto which a specific holomorphic function extends.  In one dimension, this leads to branched Riemann surfaces.  In several dimensions you also get a branched domain if you attempt to extend analytic functions .  These branched domains are called the "envelopes of holomorphy".
