B Asymptotic Expansions - General Considerations

In the previous section we demonstrated the application of asymptotic expansion techniques to obtain the high- and low-frequency limits of the velocity field for flow in a circular tube driven by an oscillatory pressure gradient. In the process, we introduced such fundamental notions as the difference between a regular and a singular asymptotic expansion and, in the latter case, the concept of matching of the asymptotic approximations that are valid in different parts of the domain. However, all of the presentation was ad hoc, without the benefit of any formal introduction to the properties of asymptotic expansions. The present section is intended to provide at least a partial remedy for that shortcoming. We note, however, 

To do this, let us consider a function, say, T(x e), that depends on spatial position and on a dimensionless parameter s that we may assume to be arbitrarily small. In the context of problems to be considered in this text, this function will usually be defined by a DE and boundary conditions, and the parameter e then appears as a dimensionless parameter in either the equation or boundary conditions. We suppose, for purpose of discussion, that T has an asymptotic expansion for small s. The general form of such an expansion is 

A very important property of an asymptotic expansion is the manner in which it converges to the function that it is intended to represent. Two facts can be stated that relate intimately to the nature of the convergence of an asymptotic expansion. First, if a function such as T(x e) has an asymptotic expansion for small e (either for all x or at least in some subdomain of x), then this expansion is unique (at least in the subdomain). However, second, more than one function T may have the same asymptotic representation through any finite number of terms. The second of these statements implies that one cannot sum an asymptotic expansion to find a unique function T(x e) as would be possible (in the domain of convergence) with a normal power-series representation of a function. This distinction between an asymptotic and infinite-series representation is reflected in a more formal statement of the convergence properties of both an infinite series and an asymptotic expansion. In the case of an infinite-series representation of some function T(x e), namely, 

according to (4-34), the difference between T and its asymptotic expansion can be made arbitrarily small for any fixed N by taking the limit e 0. It is very important to recognize that asymptotic convergence does not imply that a better approximation will be achieved by taking more terms for any fixed e, even if e is small. Indeed, it is possible that the difference between Tand its asymptotic expansion may actually diverge as we add more terms while holding e fixed. 

In this second formula, neither the gauge functions nor the spatially dependent coefficients are normally the same as those appearing in the first representation of T. Furthermore, the spatial variable x will frequently be scaled differently from its nondimensionalization in the portion of the domain where 

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