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By H. Majima

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**Example text**

If u is strongly asymptotically developable to ( [H)p and to O, then w can be taken as a (q-l)-form strongly OM ^ m asymptotically developable to (@M~H) p! and to O, respectively. 45 For this purpose, it suffices to prove the following Lemma. Let k be a positive integer inferior or equal to n and let u be of the form u = ~#1=q,lC [l,k]fldXl " Then~ there exists a (q-l)-form F holomorphic and strongly asymptotically ble in S such that u - d F = ~ # 1 = q , i C [l,k-1]gldXl" developa- If u is strongly asymptotically de- velopable to (~MiH) p and to O, then F is so t respectively.

S(c,r)) and strictly strongly asymptotically developable there, is closed with respect to the fundamental operations except the differentiation. Moreover, each fundamental operation is commutative with the operation FAj for any non-empty subset J of [l,n]. Let f be a function in an open polysector S(c,r). According to the above notations, f is strongly asymptotically developable as x tends to H in S(c,r), if 27 and only if there exists a family of functions F = {f(xl;Pj): ~ J C [ l , n ] , I=J c, pj~ ~J} such that each function f(xi;Pj) is holomorphic in S I for any non-empty proper subset J of [l,n] and any pjENJ, f(x~;p[l,n]) is constant for any p[l,n]e ~n, and for any closed (resp.

We can prove the following theorem (cf. 2). 1. For any consistent family F in S(c,r) and for any proper open subpolysector S(c',r')~ there exists a function f holomorphic and stronRly asymptotically developable in S(c',r') and TA(f) coincides with F. Let f be strongly asymptotically developable in S(c,r) and TA(f) = {f(xl,X[n,,+l,n];qj)}. j w i t h c o e f f i c i e n t s of h o l o m o r p h i c f u n c t i o n s i n l i ~ j D ( r j ) . i d e a l of H and put OMiH = p r o j o l i m k _ ~ o o ~ / ( I H ) k .