Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

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A function which has no other singularities in a given region
or in the neighborhood of a given point than non-essential ones
is said to be meromorphic in the region or in the point.
~ 3. ESSENTIAL SINGULARITIES
An analytic function of a single complex variable z may have
an isolated essential singularity, z = a, of either one of two kinds:
(a) the function may be analytic throughout the complete
neighborhood of the point a except at the point itself, and there
neither remain finite nor become infinite; (b) the function may
have poles that cluster about the point a, being analytic at all
other points of the neighborhood distinct from a.
It follows from the first theorem of ~ 1 that the first case has
nothing corresponding to it when we pass to functions of several
complex variables. But may not the second case be realized?
May not a function of several variables, f(zi,..., n), be analytic
except for non-essential singularities throughout the whole
neighborhood of a point (ai,..*, an), this point alone being
excepted? Weierstrass believed apparently that it can, for he
stated the following theorem.*
To an arbitrary continuum in the 2n-dimensional space of the
variables (zl,.., Zn) there correspond functions analytic or
having at most non-essential singularities, but having in every
boundary point a singularity of higher order.
This theorem, however, is false, as was shown by E. E. Levit
in a notable paper published three years ago, to which we shall
return later, ~~ 8, 9. In particular, it appears that an isolated
essential singularity is impossible.
~ 4. REMOVABLE SINGULARITIES
In his inaugural dissertation Riemannt stated and proved the
theorem whose practical value is so well known, namely, that
* Journ. fir Math., 89 (1880), p. 5 =Werke, 2, p. 129.
t Ann. di. mat. (3), 17 (1910), p. 61. Levi's paper appeared while a paper
of Hartogs, Math. Ann., 70 (1911), p. 217, overlapping to some extent Levi's
paper and showing in particular the impossibility of an isolated essential
singularity, was in press.
t G6ttingen Dissertation, 1851, ~ 12, =Werke, p. 23.