By Adelina Georgescu (auth.)

ISBN-10: 0412558602

ISBN-13: 9780412558603

ISBN-10: 1489945350

ISBN-13: 9781489945358

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1 = 0, P; 2 + Pf1 = 0, 2p; 1P; 2 + P; 3 = 0, ... , whence Pit = 1, P; 2 = - 1, Pi3 = 2, ... , and this corresponds to a single asymptotic expansion p 1 "' 1- e + 2e 2. So this algorithm does not yield the asymptotic behaviour of the two roots P; = [- 1 ± (1 + 4e) 112 ]/(2e). As lim,__. 0 p 1 = 1 and lim,__. 0 p 2 = oo it follows that the expansion describes only the behaviour of p 1, which is of order 1 as e--+ 0. In order to estimate the order of p2 we notice that Pz 1I. m -=- 1 e--+o 11 e Therefore pz(e) = 0 8 (1/e) as e --+0 in IR; hence p2 (e) "'p 21 e- 1 + o(e- 1) as B --+0.

Then L;a;6;(z) will necessarily be the asymptotic expansion of every function g asymptotically equal, with respect to the asymptotic sequence {6J This allows us to define the sum of an arbitrary series as being the equivalence class (with respect to the equivalence relation corresponding to asymptotic equality with respect to an asymptotic sequence) of all functions! having the given series as asymptotic expansion. The asymptotic series representing the asymptotic expansion of a functionfmay be convergent, but its limit is not necessarilyf Instead, it may be another function asymptotically equal tof For instance, the asymptotic expansion L:( -1t z-• of (1 + e-z)(l + z)- 1 as z--+0 is a series converging towards the function (1 + z) - 1 (which belongs to the sum of this asymptotic expansion).

It is seen that when the argument increases continuously, as the lines arg z = 2pn or arg z = 2(p + l)n 1 are crossed, c and c2 are subjected to jumps, and therefore the leading term of the asymptotic expansion of J v has discontinuities on these Stokes lines. However, it must be mentioned that the Bessel functions are continuous across these lines; only their asymptotic expansions, which approximate them, have discontinuities (c 1 will have a jump when the function of which it is a factor may be neglected with respect to the other; a similar situation takes place for c2 ).

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Asymptotic Treatment of Differential Equations by Adelina Georgescu (auth.)

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