Isolated Singularities and Series Expansions
Part 2: Isolated Singularities
A function f has an isolated
singularity at z0 if f is defined and differentiable
at each point of a disk centered at z0 except at the point
z0 itself.
Here are the definitions of three
functions, each with an isolated singularity at 0:
- f1(z) = sin(z)/z;
- f2(z) = cosh(z)/z;
- f3(z) = exp(1/z).
A function f has a removable singularity at a point z0 if f may be defined at z0 in such a way that the new function is differentiable at z0.
- The point 0 is a removable singularity of f1. Find the first seven terms of the series expansion of f1 about z = 0. Why is this singularity "removable"?
We will explore the behavior of these
three functions near the singularity at 0 by examining how the magnitude
|f(z)| varies over small circles centered at 0.
- Use the commands in your worksheet
to plot |f1(z)| over circles of radii 1, 0.8,
0.6, 0.4, 0.2, and 0.1. Describe how |f1(z)|
varies as z approaches 0.
A function f has a pole
of order n at z0, where n is a positive integer,
if
(z - z0)n f(z)
has a removable singularity at
z0, but
(z - z0)(n - 1) f(z)
does not.
- Find the first seven terms of
the (Laurent) series expansion of f2 about z = 0.
The point z = 0 is a pole of order one. How can you
tell it is a pole? How can you tell it is order one?
- Plot |f2(z)|
over circles of radii 1, 0.8, 0.6, 0.4, 0.2,
and 0.1. Describe how |f2(z)| varies as z
approaches 0.
- Define a function g that
has a pole of order two at 0. Find the first seven terms of the series
expansion of g about z = 0. Explain how you know that
0 is a pole of g. Explain how you know it has order two.
- Plot |g(z)| over circles
of radii 1, 0.8, 0.6, 0.4, 0.2, and 0.1.
Describe how |g(z)| varies as z approaches 0.
An isolated singularity that is not
removable and not a pole is called an essential singularity.
- Ask your computer algebra system
to find the first seven terms of the series expansion of f3
about z = 0. What happens?
- What is the Laurent series
expansion of f3 about z = 0? (This you must do on
your own. The series command will not help.) How can you tell that 0
is an essential singularity of f3?
- Plot |f3(z)|
over circles of radii 1, 0.8, 0.6, 0.4, 0.2,
and 0.1. Describe how |f3(z)| varies as z
approaches 0.
