Asymptotic expansion

First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion. If \varphi_n is a sequence of continuous functions of asymptotic variable x on some domain, and if L is a limit point of the domain, then the sequence constitutes an asymptotic scale if for every , : \varphi_{n+1}(x) = o(\varphi_n(x)) \quad (x \to L)\ . Commonly, L is taken to be zero or infinity. In other words, a sequence of functions is an asymptotic scale if each function in the sequence grows strictly slower (in the limit ) than the preceding function. If f is a continuous function on the domain of the asymptotic scale, then has a standard asymptotic expansion of order N with respect to the scale as a formal series : \sum_{n=0}^N a_n \varphi_{n}(x) if : f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = O(\varphi_{N}(x)) \quad (x \to L) or the weaker condition : f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = o(\varphi_{N-1}(x)) \quad (x \to L)\ is satisfied. Here, o is the little o notation. If one or the other holds for all , then we write : f(x) \sim \sum_{n=0}^\infty a_n \varphi_n(x) \quad (x \to L)\ . In contrast to a convergent series for , wherein the series converges for any fixed x in the limit , one can think of the asymptotic series as converging for fixed N in the limit x \to L (with L possibly infinite). == Examples ==

• Gamma function (Stirling's approximation) \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 x e^x E_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \to \infty) • Logarithmic integral \operatorname{li}(x) \sim \frac{x}{\ln x} \sum_{k=0}^{\infty} \frac{k!}{(\ln x)^k} • Riemann zeta function \zeta(s) \sim \sum_{n=1}^{N}n^{-s} + \frac{N^{1-s}}{s-1} - \frac{N^{-s}}{2} + N^{-s} \sum_{m=1}^\infty \frac{B_{2m} s^{\overline{2m-1}}}{(2m)! N^{2m-1}}where B_{2m} are Bernoulli numbers and s^{\overline{2m-1}} is a rising factorial. This expansion is valid for all complex and is often used to compute the zeta function by using a large enough value of , for instance . • Error function \sqrt{\pi}x e^{x^2}{\rm erfc}(x) \sim 1+\sum_{n=1}^\infty (-1)^n \frac{(2n-1)!!}{(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. Thus, for example, one may start with the ordinary 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 . 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, after the substitution u=w/t on the right hand side. The integral on the left hand side, understood as a Cauchy principal value, can be expressed in terms of the exponential integral. The integral on the right hand side 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 . However, by 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} \left (\tfrac{1}{t} \right ) for sufficiently small . Substituting x=-\tfrac{1}{t} and noting that \operatorname{Ei}(x)=-E_1(-x) results in the asymptotic expansion given earlier in this article. Integration by parts Using integration by parts, we can obtain an explicit formula\operatorname{Ei}(z) = \frac{e^{z}} {z} \left (\sum _{k=0}^{n} \frac{k!} {z^{k}} + e_{n}(z)\right), \quad e_{n}(z) \equiv (n + 1)!\ ze^{-z}\int _{ -\infty }^{z} \frac{e^{t}} {t^{n+2}}\,dtFor any fixed z, the absolute value of the error term |e_n(z)| decreases, then increases. The minimum occurs at , at which point \textstyle \vert e_{n}(z)\vert \leq \sqrt{\frac{2\pi } {\vert z\vert }}e^{-\vert z\vert }. This bound is said to be "asymptotics beyond all orders". == Properties ==

Uniqueness for a given asymptotic scale For a given asymptotic scale \{\varphi_n(x)\} the asymptotic expansion of function f(x) is unique. That is the coefficients \{a_n\} are uniquely determined in the following way: \begin{align} a_0 &= \lim_{x \to L} \frac{f(x)}{\varphi_0(x)} \\ a_1 &= \lim_{x \to L} \frac{f(x) - a_0 \varphi_0(x)} {\varphi_1(x)} \\ & \;\;\vdots \\ a_N &= \lim_{x \to L} \frac {f(x) - \sum_{n=0}^{N-1} a_n \varphi_n(x)} {\varphi_N(x)} \end{align} where L is the limit point of this asymptotic expansion (may be ). Non-uniqueness for a given function A given function f(x) may have many asymptotic expansions (each with a different asymptotic scale). Subdominance An asymptotic expansion may be an asymptotic expansion to more than one function. == See also ==