Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

177

~ 10. CONCERNING THE BOUNDARY OF THE DOMAIN OF
DEFINITION OF f(x, y)
Let 2 be a simple regular 3-dimensional manifold of 4-dimensional space. Then I can be represented analytically by the
equation
((Xi, x2, Yl, y2) = 0,
where x = X1 + ix2, y = Y1 + iy2; where, furthermore, o is
continuous together with its first partial derivatives; and where,
finally, not all of these four derivatives vanish simultaneously.
We will restrict ourselves to such manifolds 2 as correspond to
functions <p having continuous second derivatives.
Levi raises the question: Can a given manifold of the above
description, or a restricted piece of it, serve as part of the boundary
of a region in which a function f(x, y) is meromorphic, but
beyond which f(x, y) cannot be continued meromorphically
across any part of 2? In other words, can 2, or a piece of A,
be a natural boundary?
He finds that the answer is, in general, negative; since, for
it to turn out affirmative, so must satisfy the following necessary
condition. Let op > 0 on the side of 2 where f(x, y) is to be meromorphic. Denote by C(<) the following expression:
() = tA2f. A1+ + A2// A*1 -2 ( a + A a
(O)=2'. z —2 'ax -2 a ax ay1 ax ax2ay
X J y _x y12/ s- 2, f p ( a2 - a2(
X Vdays X Z20Y2}- 2 1 Y2- (D2 1Y xlVY2 zX20y/
where
(0 \2 (p \2 a2 52
and where Ai"P', A2"p denote similar expressions in yi, y2. Then
must C(So) < 0 in all points of Z.
If sp < 0 on the side of 2 where f(x, y) is meromorphic, then
must (fo) > 0 in all points of A.
From this result it follows that if there is to exist a function