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The simplest example, when considering a function f(n), is when there is a need to describe its properties as n becomes very large. Thus, if f(n) = n2+3n, the term 3n becomes insignificant compared to n2, when n is very large. The function f(n) is said to be "asymptotically equivalent to n2 as n → ∞", and this is written symbolically as f(n) ~ n2.
Formally, given functions f and g of a natural number variable n, one defines a binary relation
if and only if (according to Erdelyi, 1956)
This relation is an equivalence relation on the set of functions of n. The equivalence class of f informally consists of all functions g which are approximately equal to f in a relative sense, in the limit.
An asymptotic expansion of a function f(x) is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for f. The idea is that successive terms provide an increasingly accurate description of the order of growth of f. An example is Stirling's approximation.
In symbols, it means we have
The requirement that the successive sums improve the approximation may then be expressed as
In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. However, this optimal partial sum will usually have more terms as the argument approaches the limit value.
Asymptotic expansions typically arise in the approximation of certain integrals (Laplace's method, saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). The famous Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge.
Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena. An illustrative example is the derivation of the boundary layer equations from the full Navier-Stokes equations governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter, ε: in the boundary layer case, this is the nondimensional ratio of the boundary layer thickness to a typical lengthscale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often centre around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.
The method of dominant balance is used to determine the asymptotic behavior of solutions to an ODE without fully solving it. The process is iterative, in that the result obtained by performing the method once can be used as input when the method is repeated, to obtain as many terms in the asymptotic expansion as desired.
The process goes as follows:
Example. For arbitrary constants c and a, consider
This differential equation cannot be solved exactly. However, it may be useful to know how the solutions behave for large x. Start by assuming as x → ∞; we do this with the benefit of hindsight, to make things quicker. Since we mostly care about the behavior of y in the large x limit, we change variables to y = exp(S(x)), and re-express the ODE in terms of S(x),
Now suppose first that a solution to this ODE satisfies
as x → ∞, so that
as x → ∞. Obtain then the dominant asymptotic behaviour by setting
If satisfies the above asymptotic conditions, then the above assumption is consistent. The terms we dropped will indeed have been negligible with respect to the ones we kept.
is not a solution to the ODE for S, but it represents the dominant asymptotic behaviour, which is what we are interested in. Check that this choice for is consistent,
Everything is indeed consistent.
Thus the dominant asymptotic behaviour of a solution to our ODE has been found,
By convention, the full asymptotic series is written as
so to get at least the first term of this series we have to take a further step to see if there is a power of x out the front.
We proceed by introducing a new subleading dependent variable,
and then seek asymptotic solutions for C(x). Substituting into the above ODE for S(x) we find
Repeating the same process as before, we keep C' and (c-a)/x to find that
The leading asymptotic behaviour is then