1.1. FORMAL DISTRIBUTIONS
15
W
Z=W
We obtain two embeddings
(1.1.9) C((*))(M) ^ C((z,w)) «- C((w))((z)).
For example, d(£ —w)_ and S(z w)+ are the images of G C((z,w)) under
z w
these embeddings.
Now observe that both C((z))((w)) and C((w))((z)) are embedded into the
space
C[[z±1,'w;:±:1]]
of formal power series in two variables. If we start with a Lau-
rent polynomial f(zw) C((z,tt;)) (i.e., a finite sum of monomials in z±1^w±1),
then its images in C f ^ 1 , ^ 1 ] ] through the two embeddings will coincide. But
this is not true for a general element of C((z,w)). For instance, 5(z w)- and
—5(z w)+ are clearly different elements of C f ^ 1 , ^ 1 ] ] , even though they come
from the same element of C((z,w)). The difference between them is our
z w
formal delta-function 8{z w).
In fact,
c((zMw))nC{(wMz)) = C[[zM][*-\w-l\
(see the picture below), so any polar term other than z~l and w~l (such as )
will have different expansions in C((z))((w)) and C((w))((z)).
However, if we multiply by (z w), we obtain a (finite) polynomial,
namely 1. Hence if we multiply both 6(z w)- and S(z w)+ by (z w), we
obtain the same element 1 C[[2;±1,iu:t1]]. Therefore (z w)8(z w) = 0 (as we
expect from a delta-function).
Previous Page Next Page