Laurent Expansion

Date: 02/25/99 at 01:03:03
From: Genni J. & Chris N.
Subject: Laurent Expansions

Could you explain to us exactly what a Laurent Expansion is and give a 
few examples? We would appreciate if you could mention all the details.

Thank you immensely,
Genni and Chris

Date: 02/25/99 at 09:35:53
From: Doctor Rob
Subject: Re: Laurent Expansions

A Laurent series is an expression of this form:

   a(n)*x^n + a(n-1)*x^(n-1) + ... + a(0) + a(-1)/x + a(-2)/x^2 + ...,

where the a(i)'s are constants. It is an infinite sum, consisting of 
only finite non-zero terms with positive exponents of x, and 
potentially infinite non-zero terms with negative exponents of x. The 
constant coefficients can be from any field, and are often the rational 
numbers, the real numbers, or the complex numbers.

These are analogues of decimal expansions of real numbers, which have 
finitely many digits to the left of the decimal point, and potentially 
infinitely many nonzero digits to the right of it.

As an example one can find the Laurent series expansion of any rational 
function (quotient of two polynomials) by long division:

   f(x) = (x^3-2*x)/(x+3).

           x^2   - 3*x + 7 - 21/x + 63/x^2 - 189/x^2 + ...
         -------------------------------------------------------
   x + 3 ) x^3         - 2*x
           x^3 + 3*x^2
           -----------------
                -3*x^2 - 2*x
                -3*x^2 - 9*x
                -----------------
                         7*x
                         7*x + 21
                         ----------------
                              -21
                              -21 - 63/x
                               -------------------
                                    63/x
                                    63/x + 189/x^2
                                    --------------------
                                          -189/x^2
                                             ...
So,

   (x^3-2*x)/(x+3) = x^2 - 3*x + 7 - 21/x + 63/x^2 - 189/x^2 + ...

Usually these series are considered as formal objects; that is, no
convergence properties have to be taken into account, although one
does encounter convergence properties at some point when applying them 
to physical situations. These series can be manipulated in the same way 
that polynomials are manipulated, i.e. adding two Laurent series being 
done term-by-term, and the same for subtracting. Multiplication is done 
using the Distributive Law, and division using long division as is done 
above.

The polynomial part of a Laurent series is the sum of all the terms 
with non-negative exponents of x. This is the analogue of the integer 
part of a decimal real number. In the above example, the polynomial 
part is x^2 - 3*x + 7.

The idea of a degree of a Laurent series makes sense, being the largest 
exponent of x in any term. In the above example, the degree of f(x) is 
2. Such degrees can be negative.

The above Laurent series are expansions about x = 0. There also exist 
Laurent expansions about x = c, for any constant c. These take the form

   a(n)*(x-c)^n + ... + a(0) + a(-1)/(x-c) + a(-2)/(x-c)^2 + ...

You get this from the form above by substituting x-c for x everywhere.

A Laurent series expansion of a function about c will not make sense 
unless there is an integer n (which is the degree) such that the limit 
as x grows without bound of f(x)/(x-c)^n equals a nonzero constant 
(which is just a(n)). For example, f(x) = sqrt(x) will not have a 
sensible Laurent series expansion about 0, since sqrt(x)/x^n has limit 
0 if n >= 1, and has limit infinity if n <= 0.

Not all Laurent series represent rational functions. It turns out that 
the rational functions are the ones whose Laurent series coefficients 
satisfy a linear recurrence relation.

Here is another example:

   x*sin(1/x) = 1 - (1/3!)/x^2 + (1/5!)/x^4 - (1/7!)/x^6 + ...

This is definitely not a rational function. It does have a Laurent 
series expansion about 0 of degree 0.

If you have more specific questions, please write back!

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/