By Steven G. Krantz

This publication treats the topic of analytic features of 1 or extra actual variables utilizing, virtually completely, the options of actual research. This process dramatically alters the normal development of principles and brings formerly missed arguments to the fore. the 1st bankruptcy calls for just a historical past in calculus; the therapy is almost self-contained. because the e-book progresses, the reader is brought to extra subtle issues requiring extra heritage and perseverance. whilst actually complex issues are reached, the ebook shifts to a extra expository mode, with pursuits of introducing the reader to the theorems, offering context and examples, and indicating assets within the literature.

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Proof: ( 1 2 ) Let f be the function the existence of which is guaranteed by the definition. For i = 1,.. ,rn and u E U set af vi (u)= -( a ) . du; Let u0 be such that f (ao)= p. Then the set of vectors {vl (u,), . . ,v,(u,)) is linearly independent and can be enlarged to a basis for Rn by the addition of vectors %+l, . ,v,. *,Wn-m) n-m By construction DF(u,, 0) is non-singular, and the Inverse Function Theorem may be applied to obtain ( 2 ) . It is trivial t o see that ( 2 3 3), while (3 a 1)follows from the Implicit Function Theorem.

U n V n {x : g(x) # 0). CXAPTER 1. ELEMENTARY PROPERTIES Let v be a multi-index. 3 Let f be a real analytic function defined on an open subset U C Rm. Then f is continuous and has continuous, real analytic partial derivatives of all orders. Further, the indefinite integral o f f with respect to any variable is real analytic. Proof: Let f be represented near a by the power series We can choose T > 0 such that the series converges at a + t , where t = (T,T , . . ,T) E Rm. But then we see that there is a constant C such that la,l~I'I 5 C holds.

6, ( a ) )is an orthonormal basis for NSftul, (iii) each &(u)is a real analytic function of u. Let F : U x W-* 4 Rn be defined by n-m + F(u,W ) = f ( u ) wkek, n-m U E U ,W = ( W ~ , . - , W ~ - E ~ )R Of course, DF(u,, 0) is non-singular, so the Inverse Function Theorem may be applied. We conclude that the map # = f 0 lI o F-', where II is projection on the first factor, is real analytic. Note that in a sufficiently small neighborhood of p, # coincides with the "nearest point" retraction. Since there is no difficulty in extending the nearest point retraction to other points of S, we obtain the desired real analytic I retraction.