By Ivar Ekeland, Roger Témam

ISBN-10: 204007368X

ISBN-13: 9782040073688

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**Example text**

For instance, if a function is differentiable and its asymptotic expansion is derived with respect to a power sequence, then the asymptotic expansion of the derivative of that function is obtained by formal differentiation of its asymptotic expansion [36, 38]. 4 How to find an asymptotic sequence with respect to which to develop a given function In order to find the asymptotic expansion of a function we must first find the asymptotic sequence with respect to which we can make the expansion. If we know one sequence {b;} then we know a class of them ({big;}, where gi(z)=Ord(1) as z--+z 0 or {bi+g;} where gi«bi as z--+ z 0 ).

Similarly if a 1 = oo it means that the order of c5 1(z) is less than that of the functionf(z) and therefore we must reinvestigate more attentively the order off Depending on whether or not f has a limit as z--+ z 0 we have two cases. (a) If z0 belongs to the domain of definition S off then limz~z. = 0 (1) as z--+ z 0 . In this case we take c5 1 (z) = 1. If the function is regular then its asymptotic expansion as z--+ z 0 will be a Taylor series of powers of z- z 0 . , z + z 1 , (z + z 2 f (z- z 0 ), (1 + z 3 )(z- z0 ) 2 , ...

X- 11 , x- 12 , ... as x--+ oo. This shows that, although the most frequently used sequence is the sequence of the powers of (z- z0 ), some of the terms of the sequence may not occur. In the previous example the only non-vanishing coefficients were a4 k because log (x 5 +ex)= log [xs( 1 + : 4) J=log x 5 +log ( 1 + : 4) Denoting cx- 4 = u we have log(1 + u)~ u2 u3 u4 u - - + - - - + ··· 2 3 4 as u--+ 0 in IR or C. Generally it is difficult to foresee which coefficients a; are zero and it is assumed that certain a; = 0 (i < k) while ak =1 0.

### Analyse convexe et problèmes variationnels (Etudes mathématiques) by Ivar Ekeland, Roger Témam

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