Singularities of Complex Functions
Thursday, November 07, 2013
1:56 PM
Homework 3 due Friday, November 15 at 5 PM.
From last time, we saw that if we have in hand a function that is known to be analytic over some annular
domain, then one can represent that function as a Laurent series which is convergent over that annulus and
uniformly convergent over any compact subset of that annulus. In particular, if a function is known to be
analytic over an open disc, then one can write a Taylor series representation for that function which
converges over that disc, and is uniformly convergent over any compact subset.
One interesting consequence of this theorem is it gives an alternative, and often easier way
of determining the radius of convergence of a Taylor series, other than using the root test.
In particular, a Taylor series representation of an analytic function, centered at a point
will have a radius of convergence equal to the distance to the nearest singularity (point
where analyticity fails). This property is not true for real analytic functions.
Consider the complex analytic function
Imagine we compute the Taylor series for
this function about
Because Taylor series representations are unique, any way we
get the Taylor series through power series manipulations is OK, within the radius of
convergence. So for example, expand it as a geometric series:
Radius of convergence by root test:
This simple geometric observation doesn't work on real analytic functions. Consider the realvalued function
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valued function
The real-valued Taylor series will be the same (with x instead of z), so its radius of convergence
will be the same. But the function itself has no singularities on the real axis.
In addition to analytic functions being shown to have Taylor series representations expanded
about arbitrary points of analyticity, one can show conversely that a convergent power
series (Taylor series) defines an analytic function.
That is, if we're given a power series:
and we show (by say the root test)
that this converges over a domain at least as big as
then this power
series defines an analytic function inside that domain, whose derivatives can be obtained by
differentiating the power series term by term.
Proof: By the Weierstrass M-test we know that the power series will actually converge
uniformly over any closed bounded region
if
Notice that each term in the power series is analytic.
A uniformly convergent series of analytic functions must itself be analytic
because of Morera's theorem. Therefore the power series defines an analytic
function on
but
is any number
so
we have analyticity on
The fact that the derivative of the power series is simply
the sum of the derivatives:
because one can use the Cauchy integral formula for the
derivative and interchange integrals and sums by uniform
convergence.
Between these theorems, and Cauchy's theorem, and Morera's theorem, we see that there are
three equivalent ways to characterize analyticity (holomorphicity) over a open domain:
• Existence of the complex derivative over the open domain
• The integral of the function over any closed contour in the domain is zero
• The ability to express the function as a power series anywhere within the domain.
The interaction of power series with analyticity enables an interesting procedure
called analytic continuation which can take a power series representation of a
function, with only a limited domain of convergence, and extend it to define an
analytic function over a broader domain of the complex plane. More generally
analytic continuation is a way to extend a limited domain of definition of a complex
function to a broader domain. Analytic continuation can be shown to prescribe a
unique analytic function over the broader domain so long as it remains simply
connected. But sometimes (very important in practice) the analytic continuation
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connected. But sometimes (very important in practice) the analytic continuation
leads to a multivalued function. Simple example: imagine power series expansions
of Log z.
This is the notion of monodromy.
One other consequence of note from the above developments is that Laurent series can be
added, multiplied, composed, integrated, differentiated by formal series manipulations within
their domain of convergence, justified by uniform convergence and the special consideration for
differentiation.
Singularities of Complex Functions
If singularities of complex functions are not isolated, then no particular
good theory. Isolated: the function is analytic in a punctured
neighborhood of the singularity :
Examples of non-isolated singularities:
•
has a non-isolated singularity at z=0:
Cluster singularity
You can find examples of natural barriers in the texts corresponding to
certain power series.
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But isolated singularities have a good comprehensive theory because one can develop a
Laurent series representation of the complex function at least over the punctured
neighborhood of analyticity of that singularity. And therefore one can classify isolated
singularities by the behavior of the Laurent series in this punctured neighborhood.
converges absolutely over a punctured neighborhood
and uniformly over any compact subset of this neighborhood.
Principal part of the Laurent series is
Three cases:
1. There is no principal part (
if
): removable singularity
2. The principal part has a finite number of nonzero terms: pole
3. The principal part has infinitely many nonzero terms: essential singularity
Removable singularity:
Example:
We'll show later that:
converge for
Has no principal part. But we might have been hesitant to say that
was analytic at z=0 because it looks like we're dividing by zero there.
The apparent singularity at z=0 can easily be removed by simply
defining the function to take the value at the singularity by using the
Laurent series (which just looks like a Taylor series) and plugging in the
value of z at the removable singularity. In other words, there is no
problem in simply extending the definition of the Laurent series to be a
Taylor series by continuous extension to the removable singularity
(here at z=0).
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(here at z=0).
By defining
We get an analytic function defined on the non-punctured
neighborhood and represented by the Taylor series (which is the same
as the Laurent series).
Note in particular that since the Laurent series will have a well-defined
limit at the removable singularity, a Laurent series is bounded near a
removable singularity.
Poles: Principal part of the Laurent series about a pole has the form:
N is the order of the pole
is the strength of the pole.
Examples of functions with poles:
Notice that if f(z) has a pole of order N, then
will have a
removable singularity at , and has a limit there. So, by unwinding the
defintion of continuity, etc.:
Given any
when R is the outer radius of convergence of the Laurent
series, there exist constants
such that:
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That is, at a pole, a function diverges as a negative power law with
exponent equal to the order of the pole. Actually this is fairly nice
behavior on the Riemann sphere; it implies that:
at a pole
.
Essential singularity
Two theorems that characterize how nasty functions behave near
essential singularities:
Casarati-Weierstrass theorem: At an essential singularity, one can
specify any complex value
on the extended complex plane and a
tolerance and one can find arbitrarily small neighborhoods of the
essential singularity where
.
Picard theorem: In any neighborhood of an essential singularity, the
complex function will assume every value on the complex plane, except
possibly for one value.
Removable singularity
Pole
Essential singularity
Prototype example of an essential singularity is
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at z=0.
Principal part has infinitely many negative terms. Let's see how Picard
theorem applies to this function.
Realize that, by elementary properties of exponential function:
In a neighborhood
the function will cover the
exterior of a disc of radius
One other note: What about branch points? They don't even have Laurent series
representations about them because...they're multivalued. To analyze them in the
same way requires the use of Puiseux series (fractional powers); can be done but we
won't.
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