Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

179

analytic at the point in question and taking on the value p + qi
along the surface in question,-all this, at least, in a certain
neighborhood of the given point?
He finds the answer to be affirmative and the function w to
be uniquely determined, provided the surface is not what he
calls a characteristic surface, i. e., a surface along which an analytic
function of two complex variables, which is not identically zero,
vanishes. In the case of a characteristic surface, there will in
general be no solution of the problem. Suppose, for example,
that the surface is y = 0,-and the general case of a characteristic
surface is reducible to this case. Then
w(x, 0) = p + qi,
and it is evident that p + qi must be a function of x analytic
at the given point.
If this condition is satisfied, there will be, not a single, but an
infinite number of solutions.
From these results follow at once the theorems:
If f(x, y) is analytic at a point and vanishes along a noncharacteristic surface through that point, no matter how restricted that surface may be, it vanishes identically.
If f(x, y) and sp(x, y) are both analytic at a point and take on
the same values along a non-characteristic surface through that
point, however restricted that surface may be, they are identically equal to each other.
Levi-Civita extends the foregoing theorems to functions of any
number of variables.
There is a theorem of Levi's* bearing on these characteristic
surfaces. He shows that any three-dimensional manifold
sp = 0 (~ 10), in every point of which (() = 0, is composed of a
one-parameter family of characteristic surfaces.
The theorems of these last two lectures have brought out clearly
the fact that the analytic functions of several complex variables
* Ann. di Mat. (3), 17 (1910), p. 89.