Asymptotic expansions

The calculation that we did for normal cumulative distribution function when x was large is an example of an asymptotic expansion, in which one exploits the largeness (or smallness, since then the reciprocal will be large) of a variable or parameter to obtain approximations to the solution of an equation or evaluation of an integral. This is a standard tool in classical applied mathematics (see, for example, de Bruijn (1981), Bleistein and Handelsman (1975), Bender and Orszag (1978), or Lin and Segel (1988 (1974))). As shown with our simple example, such expansions can be very powerful and give intuition about much more complicated quantities. We will use asymptotic expansions in our study of stochastic population theory in Chapter 8.

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