Colloquium publications.

172

THE MADISON COLLOQUIUM.

The theorem admits of generalization to functions of any
number of variables; Hartogs, 1. c.
Levi* has given a similar theorem for the case that f(x, y) is
meromorphic in (B, K) and also in the points (~, y), where t
lies on C and y in B'. f(x, y) will then be meromorphic in (B, B').
~ 8. LEVI's MEMOIR OF 1910
We come now to one of the most important contributions of
recent years to the theory we are discussing,-E. E. Levi's
memoir of 1910.* With the aid of two lemmas, each admitting
a simple proof, Levi establishes a fundamental theorem, from
which follows with ease a complete treatment of a number of
questions in our theory which had presented themselves during
the last decade.
Lemma 1. Let f(x, y) be analytic in the cylindrical domain
B: \x\ < h, k < ~ y\ < K;
and let f(x, y) be developed in that domain into a Laurent's series:
00
(1) f(x, y) = gn(x)yn.
71=-00
In order that the analytic continuation of f(x, y) into the
region
jxj _ h, lyl < k
may have there no other singularities than non-essential ones,
it is necessary and sufficient that there exist a system of 1 + 1
functions Ai(x), i = 0, 1, * * *,,-where I may be any positive
integer, - such that
Ao(x)gn-_(x) + Al(x) gn-l+l() + + * -i- Al(x)gn(x) = 0,
where n = - 1, -2, ***. Moreover, the functions Ai(x)
are all analytic in the circle x Ij < h, and Ao(x) does not vanish
there.t
* Ann. di Mat. (3), 17 (1910), p. 61.
t This latter function may, without loss of generality, be set equal to unity.