2

WALTER RUDIN

and the series converges absolutely. Here á runs over all multi-indices, i.e., a =

(á . . . , á ), and each OL. is a nonnegative integer; also

á

á

1

á

ð

á! = á, ! . . . á ! u; = w;t ë · · ·

Ì ;

ç

and

Da = Dall-..Dann, D. = d/dzf

We now turn to a description of the geometric Situation with which we will be

dealing.

An open cone in Rn is an open set S such that ã Å S implies ty 6 S for every

positive real number t. Á typical example of such a cone is the set

Rl={yERn:

y.0 for 1 / ô é }.

If S is an open cone in Rn, then S contains a ball which contains ç linearly inde-

pendent vectors u., · · · , u . The convexity of the ball implies that S contains every

y of the form

y =

t.u.

+ · ·· + t u

U.

0,

YY 0).

Let Ë be the linear Operator on Rn that carries uk to efc (efc has 1 for its kth coor-

dinate, while all other coordinates are 0); then Ë is invertible, and A(S) contains the

closure of R7^, except possibly for the origin. This change of variables will be useful

later.

If now S is an open cone in Rn, let V be the intersection of S with some bound-

ed open ball with center at the origin of Rn. Let Å be a nonempty open set in Rn.

Define

W+=E + iV, W- =E -iV.

So W is the set of all ae = ÷ + iy in Cn such that ÷ € Å, y £ V, whereas ÷ + iy C W~

if and only if ÷ € Å and —ã € V.

The sets W and W~ are "wedges" whose "edge" is E. (We identify Å with

Å + i0; this agrees with the usual identification of a real number with a complex num-

ber whose imaginary part is 0.) It is important that W and W" need not intersect (in

fact, in the interesting cases they don't) and that therefore W ÖEl) W~ need not be an

open subset of C n , if ç 1. To see an example of this, take ç = 2, and let V be the

first quadrant of the open unit disc.

The "continuous version" of the edge-of-the-wedge theorem can now be stated.

Theorem A. / / E, W , W~ are as above, there is an open sei Ù in Cn which

contains W \JE(jW~ and which has the following property: Every continuous complex

function f on W U Å U W~ which is holomorphic in W U W~ extends to á holomorphic

function F in Ù.