## A Primer of Real Analytic Functions by Steven G. Krantz

By Steven G. Krantz

This e-book treats the topic of analytic capabilities of 1 or extra actual variables utilizing, virtually exclusively, the concepts of actual research. This strategy dramatically alters the common development of rules and brings formerly overlooked arguments to the fore. the 1st bankruptcy calls for just a history in calculus; the remedy is sort of self-contained. because the ebook progresses, the reader is brought to extra subtle issues requiring extra history and perseverance. whilst really complicated subject matters are reached, the ebook shifts to a extra expository mode, with ambitions of introducing the reader to the theorems, supplying context and examples, and indicating assets within the literature.

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**Example text**

A differential equation of order k on R" is thus an equation of the form where It is harder t o describe the general initial data (also, called Cauchy data) for a differential equation if the data is t o be specified on a real analytic submanifold: this is the situation that we have in the general Cauchy-Kowalewsky Theorem. Let bo : S -+ Rm. Then we can seek a solution u of the differential equation which also satisfies But for a differential equation of order k we should also specify the derivatives up to order k - 1.

The original papers are [CAU, pp. 52-58]) and [KOW]. The technique used in the proof is called rnajorzzation: One sets up a problem which is already known to possess an analytic solution and uses the resulting convergent power series to show that the power series arising for the original proble~riis smaller and thus is convergent. We have used essentially this technique in previous proofk, for example, in the proof of the Inverse Function Theorem. Our discussion will follow that of Courant and Hilbert, [COU].

Set for t = 1 , 2 , . . Note that each Fe is closed. By hypothesis, we have so by the Baire Category Theorem there must be some l and some open interval I C (a,p) such that Since we can always replace I by a smaller interval around any of the points in B n I , it will be no loss of generality to also assume that the interval I has length less than or equal to min(6, &). Fix such a value of l and such an open interval I. Consider any point x E I \ B. There is some maximal open subinterval, (c, d) , of I which contains x.