Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

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morphic, it is possible to enunciate the foregoing theorem for the
case that the word meromorphic is changed throughout to
analytic, and moreover essential singularity is replaced by
singularity.
Both the second lemma and the main theorem are formulated
here more generally than in Levi's paper. Levi's proof applies,
however, to the extended theorems.
~ 9. CONTINUATION. LACUNARY SPACES
Regular Regions. We will understand by a regular region
of m-dimensional space a finite m-dimensional continuum,
together with its boundary; the latter consisting of a finite
number of simple, regular, closed, non-intersecting, (m - 1)dimensional manifolds. It is obvious that this definition can
be formulated more generally, but the above is sufficient for
our present purposes.
Theorem 1. Let f(x,y) be analytic at every point of the
boundary of a regular region, T, of the 4-dimensional space of
(x, y). Then f(x, y) admits an analytic continuation throughout
T, and the resulting function will be analytic * in T.
This theorem corresponds to Levi's Corollary I, 1. c., p. 11,
but is more general. His statement of his corollary is defective.
We will speak of the proof after taking up the proof of the next
theorem.
In particular, then, it follows from the foregoing theorem
that an analytic function of two complex variables cannot have
a finite lacunary region, around which the function is analytic.
Thus, for example, no function f(x, y) exists which is analytic
in the spherical shell bounded by the hyperspheres with centres
at the origin and of radii, r = 1 and r = 1 + 5, 8 > 0, and which
has in the former hypersphere a natural boundary.
As has already been pointed out, the theorem was stated and
proven for cylindrical regions by Hartogs.
Theorem 2. This theorem differs from Theorem 1 solely in
* Cf. II, ~ 1.