Quick question on Laurent series proof uniqueness

In summary, the proof of the uniqueness of laurent series involves integrating the function 1/z^n around a loop enclosing the origin in the complex plane. This leads to the conclusion that the integral is equal to 2πiδn-1, where δ is the Kronecker delta function.
  • #1
binbagsss
1,254
11

Homework Statement



I am looking at the wikipedia proof of uniqueness of laurent series:

https://en.wikipedia.org/wiki/Laurent_series

Homework Equations



look above or below

The Attempt at a Solution



I just don't know what the indentity used before the bottom line is, I've never seen it before, would someone kindlly explain this to me or point me to a link?

Many thanks
 
Physics news on Phys.org
  • #2
Let's integrate the function ##1/z^n## in the complex plane around a loop enclosing the origin, ##z=0##. In this loop, let ##z = R e^{i\theta}## so ##dz = iR e^{i\theta} d\theta##. So (##\theta## goes from 0 to ##2\pi##):

##\int \frac{1}{z^n} dz = \int \frac{1}{R^n e^{i n \theta}} iR e^{i\theta} d\theta##
##= \frac{i}{R^{n-1}} \int e^{(1-n) i \theta} d\theta##

If ##n## is an integer and ##n \neq 1##, then we have

##\int \frac{1}{z^n} dz = \frac{1}{R^{n-1}} \frac{1}{1-n} [e^{2\pi (1-n) i} - 1] = 0## (since ##e^{2(1-n)\pi i} = 1##)

If ##n=1##, then

##\int \frac{1}{z^n} dz = \int i d\theta = 2\pi i##

So we can summarize this:

##\int \frac{1}{z^n} dz = 2\pi i \delta_{n-1}##
 
  • Like
Likes binbagsss

1. What is a Laurent series?

A Laurent series is a representation of a complex function as an infinite sum of terms, including both positive and negative powers of the variable. It is used to extend the concept of a Taylor series to functions with singularities or poles in the complex plane.

2. Why is uniqueness important in a Laurent series proof?

Uniqueness is important in a Laurent series proof because it ensures that there is only one possible representation of a given function as a Laurent series. This allows us to confidently use the series to approximate and analyze the behavior of the function.

3. How is uniqueness proven in a Laurent series?

Uniqueness in a Laurent series can be proven by showing that the coefficients of the series are uniquely determined by the function and its singularities. This can be done by comparing the series to the Taylor series of the function, and showing that the two series are equivalent.

4. Can a function have more than one possible Laurent series?

Yes, a function can have more than one possible Laurent series representation. This can happen if the function has multiple singularities with different orders, or if the function is not analytic on a certain domain.

5. What are some applications of Laurent series?

Laurent series are used in a variety of mathematical and scientific fields, including complex analysis, physics, and engineering. They are particularly useful in approximating and understanding the behavior of functions with singularities, such as in the study of fluid dynamics and electromagnetic fields.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
964
  • Calculus and Beyond Homework Help
Replies
15
Views
3K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
4K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
3K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
476
Back
Top