Laurent Series expansions

  • Thread starter rick1138
  • Start date
  • #1
192
0

Main Question or Discussion Point

Does anyone know of any examples of the explicit calculation of the Laurent series of a complex function? Any information would be appreciated.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
955
The Laurent series is simply a power series that includes a finite number of negative powers. If a function is analytic at x= a, then its Taylor's series IS its Laurent series- there are no negative powers. If a function has an essential singularity at x= a, then it does not have a Laurent series. If a function has a pole of order n at x= a, then (x-a)nf(x) is analytic at x= a. Construct the Taylor's series for (x-a)nf(x) and multiply each term by (x-a)-n.

As simple example: f(x)= ex is analytic at x= 0 so its Laurent series there is the same as its Taylor's series: 1+ x+ (1/2)x2+ ...+ (1/n!)xn+ ...

f(x)= ex/x3 has a pole of order three at x= 0 (since x3f(x)= ex is analytic at x= 0 but no lower power will give an analytic function). It's Laurent series is
x-3(1+ x+ (1/2)x2+ (1/6)x3+ (1/24)x4+...+ (1/n!)xn+...)= x-3+ x-2+ (1/2)x-1+ (1/6)+ (1/24)x+ ...+ (1/n!)xn-3+...
 
  • #3
192
0
Excellent. Exactly what I was looking for. Thanks.
 
  • #4
1
0
Construct the Taylor's series for (x-a)nf(x) and multiply each term by (x-a)-n.
I know this was six years ago, but would you believe it is the clearest explanation of Laurent series on the internet.
 
  • #5
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
And inaccurate for essential singularities. :frown: Complex analysis permits infinitely many negative powers as well.

In pure algebra, though, they usually limit Laurent series to ones that only have finitely many negative powers.
 
  • #6
391
0
i posted a similar problem in this forum

my question about Laurent is this

let be the Taylor series [tex] f(1/x)= \sum_{n=0}^{\infty}c_{n}x^{n} [/tex] valid for |x| <1

then , if i make a change of variable [tex] x=1/y [/tex]

[tex] f(y)= \sum_{n=0}^{\infty}c_{n}y^{-n} [/tex] is a LAURENT series for the fucntion f(y) valid for |x| >1 ??
 

Related Threads on Laurent Series expansions

Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
7
Views
6K
Top