Get A Primer of Real Analytic Functions PDF

By Steven G. Krantz

ISBN-10: 0817627685

ISBN-13: 9780817627683

ISBN-10: 3764327685

ISBN-13: 9783764327682

This publication treats the topic of analytic services of 1 or extra actual variables utilizing, virtually completely, the ideas of genuine research. This process dramatically alters the typical development of principles and brings formerly ignored arguments to the fore. the 1st bankruptcy calls for just a history in calculus; the therapy is almost self-contained. because the booklet progresses, the reader is brought to extra refined themes requiring extra historical past and perseverance. while actually complex subject matters are reached, the publication shifts to a extra expository mode, with objectives of introducing the reader to the theorems, offering context and examples, and indicating assets within the literature.

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Extra resources for A Primer of Real Analytic Functions

Example text

In the third section we present some work of Whitney which will both unify and supersede that which went before. 2 Besicovitch's Theorem An old theorem of E. Bore1 is as follows (see [HORI, vol. 1 Let {aj}j",, be any sequence of real or C O V I ~ Z ~ nmbers. Then there is a C" function on the internal (-1,l) such that f ( j )(0)= j ! a j . In other words, the Taylor coefficients of a Cw function at a point may be specified at will. The next theorem, due to A. 2 Let { a j } g 0 and {bj}$, b sequences of real or complex numbers.

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. Choose 0 < p < 1, and consider x with 1xj - ajl 5 pT for j = 1, 2 .

For each t E I let 1 '('1 = lim sup,,, laj ( t ) be the radius of convergence of the power series of g at t. If for each point t E I we have p ( t ) > 0 and lim inf,,t p(x)/lx - tl > 1 then f i s real analytic on I. Due in some measure to the influence of Hardy and Littlewood, mathematicians of the period described here did not study functions of several real variables. However it is not difficult t o see that the theorem of Pringsheim and Boas also holds in JRN. ) As an intuitively appealing sufficient condition for real analyticity, Pringsheim and Boas's theorem is reminiscent of an important, but unfortunately rather obscure, "converse to Taylor's theorem" that we now record.

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A Primer of Real Analytic Functions by Steven G. Krantz


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