By H. Majima
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Extra info for Asymptotic Analysis for Integrable Connections with Irregular Singular Points
Furthermore, Malgrange  and Robba  proved that a linear ordinary differential operator has a canonical formal decomposition as an operator (cf. Ince ) and translated the fact into a decomposition theorem of differential modules. ) In the mid-1970's, the sheaf of germs of asymptotically developable functions of one variable was introduced essentially by Y. Sibuya [73, 74] and after him definitively by B. Malgrange ; they developed treatments of ordinary differential equations using the sheaf of germs of functions having asymptotic expansions.
It is clear that if FA(f) belongs to OH'(r), FAj and App N coincide with those given before respectively. Let f be a function holomorphic and strongly asymptotically developable in an open polysector S(c,r) = ~i=iS(ci,ri)n . 21) g(xl;qj;Ni) N-I Xqj=of(Xl~I';qJUJ' = X~¢J'C l(-l)#J'+l~ieJ' where I' denotes the complement of J' in I. )xj,qj, As we see it above, if a holomorphic function f is strongly asymptotically developable in S(c,r)~ then the family TA(f) is consistent there. Let S be a closed (resp.
9) AppN(f) = Z¢~jC [l,n] (_I) #J+l Zje J ~pJj=of(xl ;Pj)Xj pJ by using f(xl;Pj)'s. We denote by ~'(S(c,r)) the set of all functions which are holomorphic in the open polysector S(c,r) and respectively strongly asymptotically developable to some formal power series in OH'(r) there. The set ~'(S(c,r)) is closed with respect to the fundamental operations: addition, multiplication, differentiation and integration. Moreover, each fundamental operation is commutative with the operation FAj for any non-empty subset J of [l,n].
Asymptotic Analysis for Integrable Connections with Irregular Singular Points by H. Majima