Understanding Isolated Singularities in the Residue Theorem

In summary, the Residue Theorem requires singularities to be isolated in order for it to hold. This means that for a function f defined on a region G, if we have a disk D around z_0 where all points are singularities for f, but f is holomorphic in G-D, we can still write f as a Laurent series in G-D and use the formula for non-isolated singularities. However, the interest of the isolated case is that we can often actually calculate the residue, at least for poles. While the residue theorem is true for non-isolated singularities, it is more practically useful for isolated singularities. Additionally, to construct a Laurent series that converges in an annulus, we take one
  • #1
Silviu
624
11
Hello! Why do the singularities in the Residue Theorem must be isolated? If we have let's say a disk around ##z_0##, ##D_{[z_0,R]}## where all the points are singularities for a function ##f:G \to C## with the disk in region G, but f is holomorphic in ##G-D_{[z_0,R]}##, we can still write f as a Laurent series in ##G-D_{[z_0,R]}## and thus we still have ##\int_{\gamma}f(z)dz = 2\pi i c_{-1}##, with ##\gamma## a picewise, smooth, closed path in ##G-D_{[z_0,R]}##. I know that ##c_{-1}## is called residue only if the singularity is isolated, but letting aside the nomenclature, why can't we use the formula for non-isolated singularities?
 
Physics news on Phys.org
  • #2
If the singularity is not isolated, the Laurent series will not converge in any region around the center of expansion.
 
  • #3
FactChecker said:
If the singularity is not isolated, the Laurent series will not converge in any region around the center of expansion.
But isn't the Laurent series defined in an annulus of convergence? Which means that the inner disk doesn't matter (f can have or not infinitely many singularities in there) but only the region between the inner and outer disk?
 
  • Like
Likes FactChecker
  • #4
Indeed the residue theorem is true and sometimes stated in the more general case you mention, e.g. in the book of Henri Cartan, Prop. 2.1, page 91. As he says there however, the interest of the isolated case is that there one can often actually calculate the residue, at least for poles.
 
  • #5
mathwonk said:
Indeed the residue theorem is true and sometimes stated in the more general case you mention, e.g. in the book of Henri Cartan, Prop. 2.1, page 91. As he says there however, the interest of the isolated case is that there one can often actually calculate the residue, at least for poles.
Thank you for you reply. So it is true even for the case I mentioned, but practically it is useful for isolated singularities, right?
 
  • #6
right.
 
  • #7
I am not an expert, but I seem to recall, to construct a laurent series converging in the annulus r < |z| < s, you take one power series converging in the disc |z| < s, and another series with no constant term converging in the disc |z| < 1/r. Then you replace z with 1/z everywhere in the second series, obtaining one that converges outside the circle |z| = r. adding this "negative" power series to the first one gives a laurent series converging in the intersection of the two "discs" |z| < s, and |z| > r.
 
  • #8
Silviu said:
But isn't the Laurent series defined in an annulus of convergence? Which means that the inner disk doesn't matter (f can have or not infinitely many singularities in there) but only the region between the inner and outer disk?
Sorry. Right. I stand corrected.
 

Related to Understanding Isolated Singularities in the Residue Theorem

1. What is the Residue Theorem?

The Residue Theorem is a mathematical concept used in complex analysis to evaluate integrals of functions with isolated singularities. It states that the value of an integral around a closed curve can be calculated by summing the residues of the function at its isolated singularities inside the curve.

2. What are isolated singularities?

Isolated singularities are points in a complex function where the function is not defined or is undefined. They are called isolated because they are separated from other points by a certain distance.

3. How is the Residue Theorem used to evaluate integrals?

The Residue Theorem provides a shortcut for evaluating complex integrals by identifying and summing the residues at isolated singularities instead of directly evaluating the integral. This is often a more efficient and accurate method of evaluation.

4. What is the difference between a simple pole and a higher-order pole in the Residue Theorem?

A simple pole is an isolated singularity where the function has a first-order pole, meaning the function can be written as (a constant)/(z-a) where a is the pole. A higher-order pole is an isolated singularity where the function has a pole of order greater than one, meaning the function has a more complicated form at that point.

5. What are some real-world applications of the Residue Theorem?

The Residue Theorem has applications in many areas of science and engineering, including electrical engineering, physics, and chemistry. It is commonly used in the design and analysis of electronic circuits, the study of fluid flow and turbulence, and the calculation of energy transfer in chemical reactions.

Similar threads

  • Topology and Analysis
Replies
2
Views
917
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Topology and Analysis
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
652
  • Topology and Analysis
Replies
8
Views
3K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Replies
3
Views
362
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Topology and Analysis
Replies
7
Views
2K
  • Topology and Analysis
Replies
4
Views
2K
Back
Top