An
asymptotic expansion of a function 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 . The idea is that successive terms provide an increasingly accurate description of the order of growth of . In symbols, it means we have f \sim g_1, but also f - g_1 \sim g_2 and f - g_1 - \cdots - g_{k-1} \sim g_{k} for each fixed
k. In view of the definition of the \sim symbol, the last equation means f - (g_1 + \cdots + g_k) = o(g_k) in the
little o notation, i.e., f - (g_1 + \cdots + g_k) is much smaller than g_k. The relation f - g_1 - \cdots - g_{k-1} \sim g_{k} takes its full meaning if g_{k+1} = o(g_k) for all
k, which means the g_k form an
asymptotic scale. In that case, some authors may
abusively write f \sim g_1 + \cdots + g_k to denote the statement f - (g_1 + \cdots + g_k) = o(g_k). One should however be careful that this is not a standard use of the \sim symbol, and that it does not correspond to the definition given in . In the present situation, this relation g_{k} = o(g_{k-1}) actually follows from combining steps
k and
k−1; by subtracting f - g_1 - \cdots - g_{k-2} = g_{k-1} + o(g_{k-1}) from f - g_1 - \cdots - g_{k-2} - g_{k-1} = g_{k} + o(g_{k}), one gets g_{k} + o(g_{k})=o(g_{k-1}), i.e. g_{k} = o(g_{k-1}). 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. This optimal partial sum will usually have more terms as the argument approaches the limit value.
Examples of asymptotic expansions •
Gamma function \frac{e^x}{x^x \sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots \ (x \to \infty) •
Exponential integral xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \to \infty) •
Error function \sqrt{\pi}x e^{x^2}\operatorname{erfc}(x) \sim 1+\sum_{n=1}^\infty (-1)^n \frac{(2n-1)!!}{n!(2x^2)^n} \ (x \to \infty) where is the
double factorial.
Worked example Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the
formal power series \frac{1}{1-w}=\sum_{n=0}^\infty w^n The expression on the left is valid on the entire
complex plane w \ne 1, while the right hand side converges only for |w|. Multiplying by e^{-w/t} and integrating both sides yields \int_0^\infty \frac{e^{-\frac{w}{t}}}{1 - w} \, dw = \sum_{n=0}^\infty t^{n+1} \int_0^\infty e^{-u} u^n \, du The integral on the left hand side can be expressed in terms of the
exponential integral. The integral on the right hand side, after the substitution u=w/t, may be recognized as the
gamma function. Evaluating both, one obtains the asymptotic expansion e^{-\frac{1}{t}} \operatorname{Ei}\left(\frac{1}{t}\right) = \sum _{n=0}^\infty n! \; t^{n+1} Here, the right hand side is clearly not convergent for any non-zero value of
t. However, by keeping
t small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of \operatorname{Ei}(1/t). Substituting x = -1/t and noting that \operatorname{Ei}(x) = -E_1(-x) results in the asymptotic expansion given earlier in this article. == Asymptotic distribution ==