Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

169

(a) in every point (x1, *., xk, ak+l,..., an), where xi, i = 1,
*.., k, ranges over B, and a1, j = k 1, * *.., n, is a fixed point
in Bj;
(b) in every point (ti, *"*, k, k+l, *, Xn), where ~i, i = 1,
* * *, k, ranges over the boundary Ci of Bi and xj, j = k + 1,
* *.., n, ranges over Bj.
The function will then admit analytic continuation throughout
the cylindrical region (B1, **, Bn).
In the foregoing results is contained the remarkable theorem
that a function f(x, * * *, Xn) which is analytic in every boundary
point of a cylindrical region (B1, * *.., Bn) admits analytic continuation throughout the whole region.*
This theorem holds for the general case of any four-dimensional
region, whether cylindrical or not. Cf. ~ 9.
~ 6. APPLICATION TO THE DISTRIBUTION OF SINGULARITIES
From the main theorem of the last paragraph Hartogs deduces
the following theorem relating to the distribution of the singularities of an analytic function.
Theorem. Let f(x, y) be analytic in the points (0, y), where
0 < I y I < h, and let f have a singular point at the origin, (0, 0).
Then, to each point x' of a certain region B: x < p, will correspond at least one point y' of the region B': y \ < h, such that
f(x, y) has a singular point in (x', y').
Here, again, it is useful to picture the points to ourselves in
the plane of analytic geometry. We assume the function
f(x, y) to be analytic along that part of the y-axis which lies in
the neighborhood of the origin, this latter point alone being
excepted and the function being in fact singular there. The
conclusion is that the projections on the x-axis of the singular
points of the function which lie in the rectangle (B, B') completely cover that part of the axis which lies in this rectangle.
We must not, of course, think of the singular points as dividing
the part of the region (B, B') in which the function is considered,
* Hartogs, 1. c., p. 231.