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The portion
of the Laurent series,
involving negative powers of is called the
principal part of
at The coefficient in equation
(), turns out to play a very
special role in complex analysis.
It is given a special name: the residue of the function
In this section we will focus on the principal part to identify the isolated
singular point as one of three special types.
Poles
If the principal part of at contains at least one nonzero term but the number
of such terms is only finite, then there exists a integer such that
In this case, the isolated singular point is called a pole of order A pole of order is usually referred to as a simple pole.
Examples
Consider the functions
with an isolated singularity at
Figures 1, 2 and 3 show the enhanced phase
portraits of these functions defined
in the square and
Now from the enhanced phase portraits
we can observe that is in fact a pole which order can also be easily seen,
it is just the number of isochromatic rays of one (arbitrarily chosen)
color which meet at that point.
Thus we can claim that and have poles of order 1, 2 and 3; respectively.
To confirm this let's calculate the Laurent series representation centered
at First observe that
Thus we can see that has a simple pole. On the other hand
then has a pole of order 2.
Finally, has a pole of order 3 since
Removable singularity
When every is zero, so that
In this case, is known as a removable singular point. Note that the residue at a
removable
singular point is always zero. If we define, or possibly redefine, at so that
expansion () becomes valid throughout the entire disk
Since a power series always represents an analytic function interior to its circle of
convergence, it follows that is analytic at when it is assigned the
value there. The singularity is, therefore, removed.
Examples
Consider the functions
Figures 4, 5 and 6 show the enhanced phase portraits of these functions defined
in the square and
We notice
that has a singularity at but in this case the plot does not show
isochromatic lines meeting at that point. This indicates that the singularity
might be removable.
We can confirm this claim easily from the Laurent series representation:
In this case, when the value is assigned, becomes entire. Furthermore, we can
intuitively
observe that since is a removable singular point of
then must be analytic and bounded in some deleted neighborhood
Exercise 1: Find the Laurent series expansion for and to confirm that
they have removable singularities at
Essential singularity
If an infinite number of the coefficients in the principal part () are nonzero,
then
is said to be an essential singular point of
Examples
The function has an essential singularity at since
Figure 7 shows the enhanced portrait of in the square
and The first thing
we notice is that the behaviour of near the essential singular
point is quite irregular. Observe how the isochromatic lines, near
form infinite self-contained figure-eight shapes.
defined on
In fact, a neighborhood of intersects infinitely many isochromatic
lines of the phase portrait of one and the same colour
[Wegert, 2012, p. 181].
This fact can be appreciated intuitively by plotting the simple
phase portrait of on a smaller region, as shown in
Figure 8.
Simple phase portrait: A closer view to the essential singularity.
Another example with an essential singularity at the origin is the function
Figure 9 shows the enhanced phase portrait of in the square and
Exercise 2: Find the Laurent series expansion for to confirm
that
it has an essential singularity at
Final remark
Phase portraits are quite useful to understand
the behaviour of functions near isolated singularities.
Figures 7 and 9 indicate a rather wild behavior of these functions in
a neighborhood of essential singularities, in comparison with poles and
removable singular points.
In addition, they can be used to explore and comprehend,
from a geometric point of view,
more abstract mathematical results such as the
Great
Picard Theorem,
which tells us that any analytic function with an essential singularity at
takes on all possible complex values (with at most a single exception) infinitely
often in any neighborhood of