By D. Mumford

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I i=o ' converges at least on the interval K = (a - R, a + R). 10) show that the power series E fti)' a) (x - a)i 00 i=o J" converges to f on J n K. 11 It is interesting to note that, in the reference [TM 71], a generalization of this result is proved in which plain differentiation (in several variables) is replaced by a suitable elliptic differential operator. 10 provide a useful characterization of real analytic functions that will be applied in many of the sections that follow. 12 Let f E C°O (1) for some open interval 1.

5) 36 2. 6) is absolutely convergent for Ix I < ro and F(x,f(x)) = 0. 7) Proof. l Y)xa + E lal>0 ba,kxa yk . 11) Iba,kI < C Rlal+k holds for all multiindices a E A(N) and all k = 0, 1, .... Oxa+ lal>O,IpI>0 I0I>0 k rL. 1_2) and obtain the following recurrence relations: C" = be,0. 1). 3. The Implicit Function Theorem 37 This first recurrence allows us to solve for each cei. Next we indicate how each coefficient ca of higher index may be expressed in terms of the by, j and indices cp with index of lower order.

48). ,m, 1, H(v) = mnC (1 - R) _1 We see that a solution can be defined by (1-R)_1 W(v)=mnC y+a provided a WCR (1 a - R) 1 =a. It is routine to see that W(t) C+t and to then conclude that the solution is real analytic at the origin, as required. 8 we will return to the discussion of the CauchyKowalewsky theorem, after we have introduced the machinery needed to state and prove a very general form of that theorem. 5 47 The Inverse Function Theorem In this section, we give a proof of the multivariable inverse function theorem using the special case of the Cauchy-Kowalewsky theorem proved in the previous section.