Asymptotic Expansions and Sequences

In the presentation of an asymptotic expansion, we need not restrict ourselves to power series (1, s, e, e, etc.), such as the previous examples, but we could also use a general sequence of functions 5 such that 

The second term in the RHS of Eq. 6.38 means that the error of the result of the series truncation to N terms has the order of magnitude as the sequence 6jv+i that is. 

A function y can have many asymptotic expansions simply because there are many sets of asymptotic sequences 8 that could be selected. However, for a given asymptotic sequence, the asymptotic expansion is unique, and the coefficients yj are determined as follows. First divide Eq. 6.37 by Sq to see 

Now take the limit of the above equation when e approaches zero, and make use of the asymptotic sequence property (Eq. 6.36), so we have 

Taking its limit when e approaches zero, we have the following expression for yi 

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Over the years, users of perturbation methods have evolved a shorthand language to express ideas. This reduces repetition and allows compact illustration. We first present the gauge functions, which are used to compare the size of functions, and then we present the order concept, which is convenient in expressing the order of a function (i.e., the speed it moves when e tends small). Finally, we discuss asymptotic expansions and sequences, and the sources of nonuniformity, which cause the solution for 0 to behave differently from the base case. 

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