By H. Majima

ISBN-10: 3540133755

ISBN-13: 9783540133759

ISBN-10: 3540389318

ISBN-13: 9783540389316

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**Example text**

1. Notice that the key to the proof is to solve the system of differential equations of the first order (3/~Xk)F=f, (~/3Xh)F=O (k < h ~ n ) under the condition of complete integrability (~/~xh)f=O, Let ~ be an m - b y - m with poles at most in H. (k < h ! n ) . e. d~=~A~. Then, V defines a holomorphism of sheaves of ~ - ~ P ( ~ H ) ~ c ~ m into ~ - ~ P + I ( ~ H ) ~ C ~ m natural way. 4) V > ~-~l(~H)@¢¢m V ~ ~ - ( ~ H ) ® ¢¢m V>~-~n(~H)®Ccm ... > 0 . 5) (xi~/~xi)u = x-PiAi (x)u + v i (i=n', ....

In the following sections, in the notation of Part I, we prove existence theorems of integrable systems of partial differential equations of the first order under certain general conditions by developing Hukuhara's method [28]. In order to construct formal power series solutions of systems of differential equations, we provide analogues for systems of algebraic equations. ,x n as parameters. Xn,, e i ~ u = ai(x,u) i=l ..... n ' ~ n , 1 under certain general conditions.

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.