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).