2
WALTER RUDIN
and the series converges absolutely. Here á runs over all multi-indices, i.e., a =
. . . , 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.
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