Ogitvie 



IN IN 



n:0 n=0 



as € "* 0. 



These happen to be convergent series (if c < 1), but we can inter- 

 pret them as asymptotic series just as well. The series on the left 

 is "consistent"; the one on the right is not, because individual terms 

 have their own e substructure. 



The striving for consistency can become a religion, but it is 

 not a reliable faith. Consistency (or the lack of it) tells us nothing ■ 



about the relative accuracv of otherv/ise eauivalent asvmDtotic ■ 



about the relative accuracy of otherwise equivalent asynmptotic 

 expansions. In fact, we could define a third asymptotic series with 

 terms given by: 



gQ(€) = l/(l-c) ; g^(c) =0 for n > 0. 



This series is grossly inconsistent, but one term gives the exact 

 answer for the sum of the previous series ! Occasionally one can 

 make educated guesses about such things, replacing a few consistently 

 arranged terms by a simple, inconsistent expression having much 

 greater accuracy in practical computations. Mathematically, these 

 different asymptotic series are equivalent, and, if € is small 

 enough, they will all give the same numerical results. But we want 

 in practice to be able to use values of c that are sometimes not 

 "small enough. " 



We shall work with consistent series, for the most part. In 

 spite of such possibilities of Improvement through the use of Incon- 

 sistent series. Most newcomers to this field of analysis find that 

 there Is a considerable element of art In the application of the 

 method of matched asymptotic expansions, and I personally consider 

 that the Improvement of the expansions through the development of 

 inconsistent expansions Is the highest form of this art. Except In 

 one respect, I do not Intend to pursue the possibilities of Inconsistent 

 expansions In this paper. 



The exception that I make Is the following: Many singular 

 perturbation problems lead to asymptotic- expansion solutions of the 

 form: 



I 



N 



n=0 m=0 



,c (log e) 



where a^^ does not depend on €. We can, of course, write this 

 out In a long string of terms quite consistently arranged. However, 

 my practice will be to treat the sum: 



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