Introduction to Laurent Series
Laurent series extend Taylor series by allowing expansions of complex functions around points where the function may not be holomorphic (i.e., where singularities exist). Unlike Taylor series that include only non-negative powers, Laurent series comprise both positive and negative powers of (z - z_0), accommodating isolated singularities at z_0. For a foundational understanding, refer to Introduction to Functions of Complex Variables and Holomorphicity.
Structure of Laurent Series
- Analytic Part: Contains non-negative powers (a_n terms), similar to Taylor series.
- Principal Part: Contains negative powers (b_n terms), capturing behavior near singularities.
This dual structure enables Laurent series to represent functions with poles and essential singularities, whereas Taylor series cannot.
Finding Coefficients Using Contour Integrals
Instead of derivatives as in Taylor series, Laurent coefficients are computed via contour integrals:
-
Analytic coefficients (a_n):
[ a_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}} dz ]
-
Principal coefficients (b_n):
[ b_n = \frac{1}{2\pi i} \oint_C f(z)(z - z_0)^{n-1} dz ]
Here, C is a closed curve encircling the singularity z_0 within the annular region where f(z) is holomorphic. Understanding the role of contour integrals is crucial; see Understanding Cauchy’s Theorem and Complex Integrals Explained for more on complex integrals.
Classification of Singularities
- Analytic point: All principal part coefficients (b_n) vanish; function is holomorphic at z_0.
- Pole of order n: Exactly the first n principal coefficients are nonzero, higher ones zero; e.g., simple pole if n=1.
- Essential singularity: Infinite nonzero principal coefficients.
- Zeros: If all principal coefficients vanish and several initial analytic coefficients are zero.
The behavior near these points dictates many function properties.
Residue and Its Importance
- The coefficient b_1 of the (\frac{1}{z - z_0}) term in the principal part is called the residue of f at z_0.
- Residues are crucial in evaluating complex integrals and form the foundation of the residue theorem. Delve deeper into these concepts by reviewing Understanding Cauchy-Riemann Relations and Holomorphic Functions.
Illustrative Examples
- For (f(z) = \frac{\sin z}{z^2}), the Laurent series about zero includes a simple pole (order 1) with residue 1.
- Rational functions exhibit multiple singularities (poles) of various orders, determined by the degree of factors in the denominator.
Visualizing Poles
- Singularities appear as spikes in the magnitude surface plot of |f(z)| over the complex plane, hence the term "pole" due to their resemblance to physical poles.
Conclusion and Next Steps
Understanding Laurent series and residues prepares for the residue theorem, a fundamental tool in complex variable theory dealing with contour integrals and evaluation of integrals around singularities. Stay tuned for the next lecture focusing on this critical theorem.
welcome back to my video series on complex variables today's lecture is going to be about luron series after
which we'll transition into a really important topic called the residue theorem you can think of a luron series
as a more generalized tailor series particularly one that's used in expanding out complex functions in fact
a couple of complex variables books that I've seen group the two together in the same chapter now just as there's a
Taylor's theorem for real functions there's a Lon theorem for complex functions
I won't prove it to you but here's what it says say I have two concentric circles in the complex plane C1 is the
Inner Circle C2 is the outer one these circles don't necessarily have to be finite one could be infinitely small
just covering a single point while the other could be infinitely large it doesn't matter I'll label the region
between these two circles as R suppose I have a function f of Z that's holomorphic in R suppose also that I
have a point Z KN at the center of the Inner Circle so not in the region where the function f is necessarily
holomorphic what laon theorem says is that I can expand F of Z as a luron series around Z KN and this luron series
is composed of Z minus Z in positive powers and Z minus Z KN in negative Powers with Taylor series you only have
the polinomial part whenever you expanded a real function around some point with L Series however you also
have the rational part the part with the negative Powers this portion the one with negative Powers is also called the
principal part of the luron series while the polom part is called the analytic part of the luron series this principal
part is significant because it allows lauron series to do something tailor series can't because if I plug Z into my
equation for f of Z then provided that all these B coefficients aren't zero my f of Z because of the Z minus Z KN in
the denominator will be undefined at zal Z that means I'm capable of capturing isolated singularities in the lauron
series of a complex function and this is what separates a luron series from a tailor series because for a tailor
series you only got a polinomial and you could only Center that polom around a point where the function was continuous
and differentiable with Lon series however you can expand around that point where the function isn't defined or a
singularity now with tailor series it was possible to find the coefficients of a series by finding the derivatives of
the function at the point you were expanding around but for law series it's a bit different because instead of
derivatives you now have to use contour integrals to find the series coefficients for the coefficients in the
a series are the series corresponding to the polinomial part of the luron series the formula looks something like this
while for the B series or the principal part of the lauron series the formula looks something like this note that in
these integrals C is just a closed curve inside R that's around Z KN now let's suppose that I have a complex plane with
a function f of Z that exists over that complex plane and a point Z KN within that plane at which F of Z is singular
where it's undefined let's say I surround this point with a circle C1 so that Z is the only singular Point within
the circle C1 by Lon theorem I can write F of Z as a luron series around the singularity Z KN the nature of the luron
series I write is related to the nature of the singularity at Z so for instance if my entire principal part or the B
series is zero then F of Z is analytic at zal Z in contrast if all infinite B series coefficients are nonzero then F
of Z is said to have an essential Singularity or an essential pole at Z KN on the other hand if only the first N B
series coefficients are non zero while all the B series coefficients above the index n are zero then Z is said to be a
pole of order n a pole of order one has a special name it's called a simple pole similarly if the entire principal
part of the luron series is zero and all A's below a n are zero then Z KN is a zero of order
n finally and this is probably the most important definition out of all the ones mentioned so far the coefficient B1 in
the principal part of the lauron series expansion around Z is called the residue of f of Z at Z so that would mean this
B1 over here let's look at a simple example to illustrate these points suppose we have a complex function given
by sin Z over Z ^2 well we can recall that the tailor expansion of s is x - x Cub 3 factorial + x 5 5 factorial and so
on in that case we can deduce that the lon expansion of s Z z^2 is just the tailor expansion in terms of Z / Z
^2 thus F of Z is 1 / Z Plus Z over 3 factorial plus Z Cub over 5 factorial and so on in this case we can see that F
of Z has a simple pole at Z not equals z because there's no terms that contain 1 / z^2 or 1 Z Cub or any terms that have
higher negative powers of Z we can also see that the residue of f of Z at Z equals 0 is just one because
the coefficient of 1 / Z is just one another example is a more complicated rational function I won't
write the lon series for this because it would take really long but I will note a couple of important things one is that
there's three singular points here one at zal 8 another at zal 2 and the third at
zal1 another thing to note is that zal 8 is a pole of order 2 zal 2 is a a pole of order 1 and Z1 is a pole of order 3
how I determined the order of these poles was by looking at the exponent next to the terms corresponding to the
singularities in the denominator because you can imagine that if we tried to expand out this rational
function as a luron series say if we expand around zal1 we won't find a term that's you
know 1 over Z +1 to the^ 4 you can see why because it only goes up to Z +1 cu so hopefully you can understand what I
mean when I associate the orders of these poles with the function that I'm given now you might wonder why I keep
calling these singularities poles the reason's pretty simple but interesting if I draw a surface plot of the
magnitude of f of Z over The Complex plane let's take the second example here then over the points of Singularity F of
Z's magnitude is going to approach Infinity so the surface will look like it has a huge spike in it at the point
of the singularity since this Spike looks like a pole people decided to call the singularity a pole I'm going to end
this video here in the next video we're going to move on to what is probably the most important topic in complex
variables and that's the residue theorem
A Laurent series is an expansion of a complex function around a point that may include both non-negative and negative powers of (z - z_0), allowing it to represent functions with singularities. In contrast, a Taylor series only includes non-negative powers and applies to holomorphic points without singularities.
Laurent coefficients are calculated using contour integrals around a closed curve enclosing the singularity z_0. The analytic coefficients (a_n) are found by integrating f(z) divided by (z - z_0)^{n+1}, while the principal coefficients (b_n) involve integrating f(z) multiplied by (z - z_0)^{n-1}, each scaled by 1/(2πi).
The analytic part contains non-negative powers and represents the regular behavior of the function near the point. The principal part includes negative powers and captures the singular behavior, such as poles or essential singularities, of the function at that point.
If all principal part coefficients (b_n) are zero, the function is analytic at the point. A pole of order n corresponds to the first n principal coefficients being nonzero with higher ones zero. An essential singularity has infinitely many nonzero principal coefficients. Zeros occur when principal coefficients vanish and initial analytic coefficients are zero.
The residue is the coefficient b_1 of the 1/(z - z_0) term in the principal part of the Laurent series. It is vital because it allows the evaluation of complex integrals via the residue theorem, serving as a key tool in complex analysis for computing contour integrals around singularities.
For the function f(z) = sin(z)/z^2 around zero, the Laurent series shows a simple pole of order 1. Its residue, the coefficient of 1/(z), is 1. This example illustrates how singularities and residues appear in practice.
Poles appear as sharp spikes or peaks in the surface plot of the magnitude |f(z)| over the complex plane. These visual spikes resemble physical poles, which is the origin of the term used to describe singularities of this kind.
Heads up!
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