Download e-book for iPad: A Comprehensive Introduction to Differential Geometry by Michael Spivak

By Michael Spivak

ISBN-10: 0914098713

ISBN-13: 9780914098713

Publication through Michael Spivak, Spivak, Michael

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Example text

Compare the functions l u ( x , r)l and

11 EXAMPLES OF CLASS II We show here, as corollaries of a general theorem, that the functions k ( x , t ) and h ( x , r ) as functions of t are in the class I1 for - 0 0 < r < co. Of course they are analytic in any interval not including t = 0. 1 1. p > 0, m 2. f ( x ) = x-"'e-P/", = =$ f ( x ) E 11,, 0, -00 x x < x < 00, > 0, > 0, < 0, r < 2p. For x > 0 we express f ( x ) by Cauchy's integral formula, integrating over the circle (Figure 5), r = {Z I Z = x[l + (1 - c ) e i e ] ) . (1) 47 11. EXAMPLES OF CLASS II Flguro 5 We obtain for k = 0, 1, 2, .

3 9. OPERATIONAL CALCULUS 43 It is evident that the two definitions do not have the same domain of applicability. The first is meaningful only if f ( x ) E C" and if the series converges, the second can be applied to integrable functions for which the integral converges. We discuss later a class of functionsf(x) for which the two definitions are equivalent. As a further example of the operational calculus, let 9 stand for differentiation with respect to t and seek to assign a meaning to cosh xm.

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A Comprehensive Introduction to Differential Geometry by Michael Spivak

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