Laurent Series 219lytic at ZQ and there exists a S > 0 so that the function / is analytic in the
region {z : 0 < \z — ZQ\ < 6}.
An isolated singular point ZQ is called
- removable, if the function can be defined at zo so that the result is an
analytic function at ZQ. - a pole of order k, if k is the smallest value of the positive integer
power n with the property that function (z — zo)nf(z) has a removable
singularity at ZQ. - an essential singularity, if it is neither a removable singularity nor
a pole.
A pole of order 1 is called simple.
Without going into the details, let us mention that the point z = 0 is not an
isolated singularity of f(z) = z1//2 in the sense of the above definition, but
rather a branching point of order two. The reason is that the square root
yfz has two different values in every neighborhood of z = 0. As a result,
in any neighborhood of zero, there is no unique number assigned to z^1 ^^2
and f(z) = z^1 /^2 is not a function in the sense of our definition. Similarly,
z = 0 is a branching point of order three for the function f{z) = z-1/^3
(the fact that / is unbounded near z = 0 is not as important as the three
different values of yfz in every neighborhood of z = 0), and z — 0 is a
branching point of infinite order for the function f(z) = \nz. The study
of branching points and the related topics (multi-valued analytic functions,
Riemann surfaces, etc.) is beyond the scope of our discussions.
Note that the closed disk {z : \z — ZQ\ < Ri} can contain several
points where / is not analytic, and ZQ is not necessarily one of them.
In the special case of the Laurent series with R\ — 0, ZQ is an isolated
singular point of the function /, with no other singular points in the do-
main {z : 0 < \z — ZQ\ < R} for some R > 0. The corresponding Lau-
rent series is called the expansion of / at (or around) the isolated
singular point ZQ. This expansion has two distinct parts: the regular
part, consisting of the terms with non-negative k, J2k>ock(z ~~ zo)k, and
the principal part, consisting of the terms with the negative values of k,
2fc<o Ck(z ~ zo)k- As the names suggest, the regular part is a function that
is analytic at ZQ (this follows from Theorem 4.3.4), and the principal part
determines the type of the singularity at ZQ (this follows from the exercise
below).