singular.htm Isolated Singularities

The Laurent Expansion

Theorem. Let f(z) be analytic in the region {z | 0 < |z| < R }. Then for 0 < |z| < R,
f(z)
= ∞
∑
n = −∞ 
 an zn,
an
= 1
2πi
 ⌠
(⎜)
⌡


Cr 
 f(ζ) ζ−n − 1  dζ.
Here r is any number such that 0 < r < R.
The series
f1(z) = ∞
∑
n = 0 
 an zn
is analytic in {z | |z| < R },
The series
f1(z) = −1
∑
n = −∞ 
 an zn
is analytic in {z | 0 < |z| }.
Note that
•
If f(z) be analytic in the region {z ||z| < R }, then
an = 1
2πi
 ⌠
(⎜)
⌡


Cr 
 f(ζ) ζ−n − 1   dζ = 0, n = −1,−2, ….
•
If f(z) be analytic in the region {z |0 < |z| < R }, then
a−1 = 1
2πi
 ⌠
(⎜)
⌡


Cr 
 f(ζ)   dζ
is called the residue of f(z) at z=0.


Isolated Singularities
For the moment, we shall consider a function f(z) analytic in the punctured disk
⋅
D
 
R 
 = {z |0 < |z| ≤ R}.
Thus the possible singularity of f(z) at z=0 is isolated .
Then
f(z)
= ∞
∑
n = −∞ 
 an zn,
an
= 1
2πi
 ⌠
(⎜)
⌡


Cr 
 f(ζ) ζ−n − 1  dζ.
•
The coefficient of z−1 is called the residue of f(z) at z=0, and is written
Res(f,z=0) = Resf(z)|z=0 = 1
2πi
 ⌠
(⎜)
⌡


Cr 
 f(ζ)  dζ.
•
If f(z) = ∑n = 0∞ an zn, f(z) may be extended by defining f(0) = a0, and the resulting function is analytic in |z| ≤ R. In this case the singularity is removable .

•
If f(z) = ∑n = N∞ an zn, N ≥ 0, aM ≠ 0, f(z) is said to have a zero of order N at z=0. Near z=0,
f(z) = zN ·g(z)
, where g(z) is analytic in |z| ≤ R, g(0) ≠ 0.

•
If f(z) = ∑n = − M∞ an zn, M ≥ 0, a−M ≠ 0, f(z) is said to have a pole of order M at z=0. Near z=0,
f(z) = z−M ·g(z)
, where g(z) is analytic in |z| ≤ R, g(0) ≠ 0. The function is also meromorphic in · DR.

•
If f(z) = O(|z|−M), M ≥ 0, the preceding exercises show that a−M−1 = 0, a−M−2 = 0, …. Thus f(z) at z=0 has a pole of order at most M.

•
At z=0, f(z) has a pole of order M iff there are positive constants c1 and c2 such that
c1
|z|M
 ≤ |f(z)| ≤ c2
|z|M
 .


Isolated Essential Singularities



Definition. If f(z) = ∑n = − ∞∞ an zn, an ≠ 0 for infinitely many negative n, then f(z) is said to have an essential singularity at z=0.

Analytic functions which have isolated essential singularities behave very badly near the essential singularity.

Theorem (Little Picard). Suppose that f(z) has an essential singularity at z=0. Then for any complex number w0, in any neighborhood of z=0, f(z) gets arbitrarily close to w0.
Proof of the Little Picard Theorem: The proof is by contradiction.If there is a neighborhood · Dr = {z | 0 < |z| < r} in which f(z) − w0 is bounded away from 0, then
g(z) = 1
f(z) − w0
is analytic and bounded in · Dr. Thus g(z) has a removable singularity at z = 0 and a zero of order N, N ≥ 0. Thus g(z) = zN ·h(z), h(z) analytic near z = 0 and h(0) ≠ 0. Possibly shrinking r, we may assume that h(z) ≠ 0 in Dr = {z | |z| < r}. Then
f(z) − w0 = z−N · 1
h(z)
 .
It follows that f(z) has a pole of order at most N at z = 0.


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On 03 May 2012, 13:52.