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FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

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a single boundary point of T. This theorem has, moreover,
recently been extended to the most general Riemann's surface.*
It is clear from the foregoing that such a theorem cannot
hold for functions of more than one variable.
~ 2. NON-ESSENTIAL SINGULARITIES
The analogue of a pole of a function of a single variable is a
point (ai, * -, an), in whose neighborhood the function can be
written in the form
(1) ^ xF(z1, * n) = H(zi, - -, n)
G7(zl, z,.)
where G and H are both analytic at (a, * *, an) and
G(al, *.. an) = 0, H(ai,, an) $ 0.
Here, F becomes infinite for all methods of approach to the
point, just as in the case n = 1. We shall denote such a point
as a pole, or as a non-essentially singular point of the first kind.t
But even a rational function can have a more complicated
singularity. Suppose that G and H are polynomials relatively
prime to each other, both vanishing at (a,, I, an); e. g.,
w
F(w, z) = -- (a, a2) = (0, 0).
z
Here, the function can actually take on any arbitrarily assigned
value in a point of an arbitrarily assigned neighborhood of the
singular point in question.
We are led, then, to a second kind of singularity, the function
still being of the form (1), but H vanishing also at the point in
question, though still being prime to G. Such a point is called
a non-essentially singular point of the second kind. In the
neighborhood of such a point, which we will take as lying at the
* The question has been treated by Koebe, Freundlich, and Osgood; cf.
Osgood, Funktionentheorie, v. 1, 2d ed., 1912, p. 747.
t Weierstrass, Werke, 2, p. 156.