1.1. FORMAL DISTRIBUTIONS

13

already have Ar solutions supported on the pth diagonal: wp'bl+1dlv5(z — w),i =

0,..., N — 1. Their coefficients are polynomials in n of degree iy so these solutions

are linearly independent, and hence span the space of solutions of (1.1.7) restricted

to the pth diagonal. Therefore we obtain that the general solution of (1.1.7) is of

the form (1.1.6).

1.1.6. Remark. Recall that a module M over the ring O(X) of regular func-

tions on an affine variety X is said to be supported on a subvariety Y C X ,

if each / £ M is annihilated by some power of the ideal of functions vanishing

on y . The above lemma means that the series of the form

f(w)dxw5(z

— w),

where f(w) £

C[[K;

- 1

]]

and i € Z+, span the maximal

C[2;±1,t^±1]-submodule

of

C[[2±1,ty:t1]]

supported on the diagonal {z = w} in (C

x

)

2

.

1.1.7. Analytic interpretation. Formula (1.1.4) can also be interpreted as

follows. Let us write

(

,L8) ,(2_w)

=

i

E ( f r +

i

E(

£

r

.

n0 n0

6- S+

When z and w take complex values, such that \z\ \w\, the series S(z — w)~

converges to the meromorphic function . On the other hand, when \z\ \w\,

z — w

the series 5(z — w)+ converges to . Formula (1.1.4) means that their sum

z — w

is "supported at z = w".

To make this more precise analytically, let us set w equal to a non-zero complex

number. Then 5(z — w) becomes a formal power series in one variable z, and hence

defines a functional on C[z,

z~1].

The value of this functional on g(z) G C[z,

z~l]

equals the value of g at w, so this functional can be considered as the "delta-

function at the point w". On the other hand, 5(z — w)± also give rise to functionals

on C[z, £ - 1 ], whose values of g(z) £ C[z, z~l] are equal to

=F lim / ^g^dz.

where |g| 1. Thus, we can write

8(z — w)± = =p lim —,

in the sense of distributions. Such distributions are called the boundary values of

holomorphic functions. Then formula (1.1.8) means that S(z — w) is the difference

of boundary values of functions holomorphic in the interior and the exterior of the

circle \w\ = const (see the picture).