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Extra info for A treatise on the theory of determinants
Also obtain an expression for the remainder, as an integral, and find an estimate for it. 13 Exponential integral. Find an asymptotic expansion of as this will involve Euler’s constant It might be helpful, first, to show that 44 1. 14 An integral I. 15 An integral II. 16 An integral III. 17 An integral IV. Use an appropriate integration by parts to find an expansion, valid as of the integral in particular, find the general term and also an expression for the remainder. Show that this is an asymptotic expansion and that the series diverges.
5 Limits involving exp(x) and ln(x). 6 Examples of O, o and ~. 7 Dominant behaviour. e. find g (x) so that as for each of these functions, f(x). 8 Properties of O and o. For the limit (a) given that f (x) = O[g(x)], F(x) = O[G(x)], show that f F = O(g G); (b) given that f (x) = O[g(x)], F(x) = o[G(x)], show that f F = o(g G). 9 Asymptotic sequences. Verify that these are asymptotic sequences, where n = 0, 1, 2, . . 10 Non-uniqueness of asymptotic expansions. Find two asymptotic expansions of the function valid as based on the asymptotic sequences: (a) for a suitable function which is to be deter- mined.
Let us investigate this by returning to our previous example. 61) further, using we have no information about terms We retain terms O(1) and and smaller. 64) are identical, when expressed in terms of the same variable (x or X). 62), are said to match to this order (because we can match only the terms available in the original expansions). The matching principle is a fundamental tool in the techniques of singular perturbation theory; it is invoked, sometimes as a check, but more often as a means for determining arbitrary constants (or functions) that are generated in the solution of differential equations.