V.1.
Classification of Singularities 1
Chapter V. Singularities
V.1. Classification of Singularities
Note. In this section, we define various types of singularities of a function and
develop the idea of a Laurent series.
Definition. A function f has an isolated singularity at z = a if there is an R > 0
such that f is defined and analytic in B(a; R) \ {a}, but not in B(a; R). Point a is
a removable singularity if there is an analytic function g : B(a; R) → C such that
g(z) = f (z) for 0 < |z − a| < R.
Example. Functions f1 (z) = 1/z and f2(z) = sin z/z, and f3 (z) = exp(1/z) each
have isolated singularities at z = 0. As shown in Exercise V.1.1, f2 (z) = sin z/z
has a removable singularity.
Theorem V.1.2. If f has an isolated singularity at a then z = a is a removable
singularity if and only if lim (z − a)f (z) = 0.
z→a
Definition. If z = a is an isolated singularity of f then a is a pole of f if lim |f (z)| =
z→a
∞. If an isolated singularity is neither a pole nor a removable singularity it is called
an essential singularity.
V.1. Classification of Singularities 2
1
Example. Function f (z) = for m ∈ N has a pole at z = a. Function
(z − a)m
g(z) = exp(z −1 ) has an essential singularity at z = 0. In fact, a function with a
pole at z = a has a well defined form, as given next.
Proposition V.1.4. If G is a region with a ∈ G, and if f is analytic in G \ {a}
with a pole at z = a, then there is a positive integer m and an analytic function
g(z)
g : G → C such that f (z) = .
(z − a)m
Definition. If f has a pole at z = a and m is the smallest positive integer such
that f (z)(z − a)m has a removable singularity at z = a, then f has a pole of order
m at z = a. A pole of order 1 is called a simple pole.
Note. If f has a pole of order m at z = a, then f (z) = g(z)/(z − a)m where g is
analytic in B(a; R) (for some R > 0), so
X
∞
m−1 m
g(z) = Am + Am−1 (z − a) + · · · + A1 (z − a) + (z − a) ak (z − a)k
k=0
and
Am Am−1 A1
f (z) = + + · · · + + g1 (z)
(z − a)m (z − a)m−1 (z − a)
where g1 is analytic in B(a; R) and Am 6= 0.
V.1. Classification of Singularities 3
Definition. If f has a pole of order m at z = a and f satisfies
Am Am−1 A1
f (z) = + + · · · + + g1 (z)
(z − a)m (z − a)m−1 (z − a)
then
Am Am−1 A1
+ + · · · +
(z − a)m (z − a)m−1 (z − a)
is called the singular part of f at z = a.
Note. We will see that an essential singularity behaves rather like a pole of infinite
order. This then produces an infinite singular part. First, some definitions.
Definition V.1.10. If {zn | n ∈ Z} is a doubly infinite sequence of complex
P P∞ P∞
numbers, then ∞ a
n=−∞ n is absolutely convergent if both a
n=0 n and n=1 a−n
are absolutely convergent. If these series are absolutely convergent then define
X
∞ X
∞ X
∞
zn = z−n + zn .
n=−∞ n=1 n=0
P∞
If un is a function on a set S for n ∈ Z and un (s) is absolutely convergent for
n=−∞
P P∞
every s ∈ S, then the convergence is uniform over S if both ∞ n=0 un and n=1 u−n
converge uniformly on S.
Definition. If 0 ≤ R1 < R2 ≤ ∞ and a is any complex number, define
ann(a; R1, R2) = {z | R1 < |z − a| < R2}.
V.1. Classification of Singularities 4
Note. We now deal with a series representation of a function analytic on an
annulus.
Theorem V.1.11. Laurent Series Development.
Let f be analytic in ann(a; R1, R2). Then
X
∞
f (z) = an (z − a)n
n=−∞
where the convergence is absolute and uniform over the closure of ann(a; r1, r2 ) if
R1 < r1 < r2 < R2. The coefficients an are given by
Z
1 f (z)
an = dz (1.12)
2πi γ (z − a)n+1
where γ is the circle |z − a| = r for any r with R1 < r < R2. Moreover, this series
is unique.
Note. The proof of Theorem V.1.11 is in a, sort of, self contained supplement.
The Laurent series allows us to classify isolated singularities.
Corollary V.1.18. Let z = a be an isolated singularity of f and let f (z) =
X
∞
an (z − a)n be its Laurent expansion in ann(a; 0, R). Then
−∞
(a) z = a is a removable singularity if and only if an = 0 for n ≤ −1,
(b) a = z is a pole of order m if and only if a−m 6= 0 and an = 0 for n ≤ −(m + 1),
and
(c) z = a is an essential singularity if and only if an 6= 0 for infinitely many negative
integers n.
V.1. Classification of Singularities 5
Note. If f has an essential singularity at z = a, then limz→a |f (z)| does not exist.
The text says: “This means that as z approaches a the values of f (z) must wander
through C.” The following result shows that this wandering is very intense.
Theorem V.1.21. Casorati-Weierstrass Theorem.
If f has an essential singularity at z = a then for every δ > 0, {f (ann(a; 0, δ)}− = C.
Note. A more general result concerning the behavior of f near an essential singu-
larity is in Chapter XII (the last chapter of the text, page 300):
Great Picard Theorem.
Suppose an analytic function has an essential singularity at z = a. Then in each
neighborhood of a, f assumes each complex number, with one possible exception,
an infinite number of times.
Note. Function f (z) = exp(1/z) is such a function, and it clearly does not take
on the value 0.
Note. For the record, from page 297 we have:
Little Picard Theorem.
If f is an entire function that omits two values, then f is a constant.
Note. Of course, f (z) = ez is an example of a function omitting one value.
Revised: 3/29/2018
