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LECTURE III
SINGULAR POINTS AND ANALYTIC CONTINUATION
~ 1. INTRODUCTION
The simplest singular points which an analytic function of a
single complex variable can have are poles, isolated essential
singularities, and branch-points.
A function of several complex variables cannot have an isolated
singularity, if we except the trivial case of a removable singularity,
i. e., a singularity such that the function becomes analytic at
the point in question when a suitable value is assigned to it
there.*
For example, the function of the single variable z,
fz) = 1 -z
has an isolated singularity at the point z = 0.
But the function of the two complex variables w = u + vi,
z = x + yi:
F(w, z)= -,
has a whole two-dimensional manifold of singularities in the
four-dimensional space of these variables, namely, the points
(u, v, 0, 0).
It is a theorem due to Weierstrass and proven by Runget
that to an arbitrary continuum T of the complex z-plane there
correspond functions of z which are analytic at every point of T
and which furthermore cannot be continued analytically over
* This result can be obtained directly from Cauchy's integral formula or
Laurent's series. It was stated by Hurwitz in his Zurich address, Verh. des
1. intern. Math.-Kongresses, 1897, p. 104.
t Acta, 6 (1884), p. 229.
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