By Steven G. Krantz

This ebook treats the topic of analytic capabilities of 1 or extra genuine variables utilizing, nearly completely, the strategies of genuine research. This strategy dramatically alters the usual development of principles and brings formerly overlooked arguments to the fore. the 1st bankruptcy calls for just a heritage in calculus; the remedy is sort of self-contained. because the booklet progresses, the reader is brought to extra refined themes requiring extra historical past and perseverance. while actually complicated issues are reached, the e-book shifts to a extra expository mode, with targets of introducing the reader to the theorems, supplying context and examples, and indicating resources within the literature.

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L,O)= g i ( y l ? - * Y Y Y Z ) and where AiYnis the algebraic function of the components of an (n + 1)x ( n+ 1) matrix which gives the entry of the inverse matrix in the ith row and (n 1 ) row. ~ ~Thus we see that the component functions u1, . ,U,+I of F-' satisfy a real analytic system of partial differential equations with real analytic initial data. Further, the initial value problem is of the restrictive type dealt with in the previous section. Therefore, the functions u l , ... ,u,+l are real analytic in a neighborhood of (0,.

V)vap+vxp. CL Choose 0 < p < R. Consider We then estimate ZJ E Rm with lyj - xjl 5 plxjl for all j . 6. FUNCTIONS OF SEVERAL VARIABLES Finally we note that, for some 29 on the line segment from x to y, So we can estimate and observe that the last series approaches 0 as N approaches oo. W As our last result in this section, we show that the composition of real analytic functions is real analytic. 7 If fl, f 2 , . . )). ), 7 fm(a)), then g[fl(x), fi(x), frn(x)] is real arialytic in a neighborhood of a.

Remark: A similar result clearly holds for the right-hand endpoint of the interval [c, dl. The proof of the theorem will require a second application of the Baire Category Theorem. Proof of the Theorem: Arguing by contradiction, suppose there are a and p with a < a < p < b such that (a, p) contains a point at which f is not real analytic. Let 3 denote the set of points in [a,p] at which f is not real analytic. Then B is closed and thus may be considered in its own right as a complete metric space.