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By Steven G. Krantz

This publication treats the topic of analytic features of 1 or extra genuine variables utilizing, nearly completely, the thoughts of actual research. This strategy dramatically alters the typical development of principles and brings formerly overlooked arguments to the fore. the 1st bankruptcy calls for just a history in calculus; the therapy is almost self-contained. because the ebook progresses, the reader is brought to extra subtle themes requiring extra heritage and perseverance. whilst actually complex issues are reached, the booklet shifts to a extra expository mode, with objectives of introducing the reader to the theorems, offering context and examples, and indicating resources within the literature.

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It should be clear from the proof of Abel's Lemma that Int B c C. It is, ¤ of course, trivial that C c Int B, so we have C = Int B. 6 For a set S c Rm, we define log 11 SlI by setting The set S is said to be logarithmicallgr convex if log llSll is a convex subset of Rm. 7 Fop. a power series vergence C is logarithmimlly convex. C, a,xp, the domain of con- Proof: Fix two points y, z E C and 0 5 X 5 1. , ,. ~ m and ) t = ( z 1 , t 2 , .. , t m )NOWy E C implies y E IntB so, for some E > 0, (1gr31 E, 1y21 E , .

It was Hassler Whitney [WHI] who discovered the correct multi-variable analogue for this fact. He was able to exploit it to prove several important extension and approximation theorems. Even today Whit ney b theorems, and especially his techniques, exert a decisive influence over the directions that real analysis has taken. 1 (Whitney Decomposition) Let 51 E lRN be an open set. Then there are closed cubes Qk such that 3. For each k,diam (Qk) -< dist (Qk,'51) 5 4 diam (Qk). In what follows, when Q C W N is a cube with center xo and c > 0 we let cQ denote the set {x E W N : x0 (l/c)(x - xO)E Q).

Proof: We may and shall assume that a is the origin in Elk and that 0 = f l (0) = f 2 (0) = . . = fm (0). We can choose E > 0 such that the open ball of radius E about the origin in Rm is contained in the domain of convergence of the power series representing g. Since each f j is continuous we can choose an E' > 0 such that the open ball about the origin in EXk is contained in the domain of each f j and f, maps the open ball of radius E' into the open interval of radius € 1 6 . Now, consider an arbitrary x E IRk which is in the open ball of radius E' and is also in the domain of convergence of the power series representing f, at the origin, for all j.

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