By John Wermer (auth.)

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**Extra info for Banach Algebras and Several Complex Variables**

**Example text**

Note n o u = identity. Step 2. Find a a-closed form <1>1 defined in a neighborhood of r-: I(PI' . , Pr) with <1>1 = ¢(n) on L . Step 3. By induction hypothesis, 3\1l in a neighborhood of t-: 1 (PI> . . ,Pr) with (3\1l = <1>1' Put IjJ = \Il(u). Then 81jJ = (8\1l)(u) = * = (A ' ¢)(n), defined = 0 outside n -I(O). Let X be a form of type (P, q) defined in a neighborhood of P": l(Pl " . *

Put n = Ux€KB(x, r/2). For' E n, define F(O = F x(,) if' E B(x, r/2), x E K . To see that this value is independent of x E K , suppose that' E B(x, r/2) (') B(y, r/2), x, y E K . Then y E B(x, r) Il K. Hence Fx(Y) = H(y). Also, Fy(Y) = H(y). Hence F; and F; are two holomorphic functions in B(x, r) Il B(y, r/2) with F; = F; = Zl there and F x = F y at y. So F),) = Fy{O. ) Thus FE H(n) and F 2 = Z1 in n. D . 4 holds when the square-root function is replaced by anyone of a large class of multivalued analytic functions.

5. J I ,J blJ dZ I 1\ dz]. J dZ k 1\ dZ I 1\ dzJ . 3 there exists a neighborhood AlJ E COO(nt> with V. :in and, for each I and J, k> Put WI = I I ,J - aWl AlJ dZ I 1\ dZJ " = L. I ,J,k = E Ap,q-I(n l) , oAlJ _ Y dZ k 1\ dZ I _ 1\ dZ J Zk L alJ di ; 1\ dZ I V. J where y is a form of level ~ v - 1. Thus aWl = dz; 1\ r:t. + y, Hence aWl - is a form of level ~ v-I. W = Y - 13 Also a(y - 13) = a(aw l - w) = o. q - l(Q 2) with = y - fJ. Then rE ar a(w l - r) = aWl Wl - r is now the desired w*. ar = W + (y - fJ) - (y - fJ) Q2 of d" and = w.