Inmathematics, anasymptotic expansion,asymptotic series orPoincaré expansion (afterHenri Poincaré) is aformal series of functions which has the property thattruncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations byDingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

The theory of asymptotic series was created by Poincaré (and independently byStieltjes) in 1886.^[1]

The most common type of asymptotic expansion is apower series in either positive or negative powers. Methods of generating such expansions include theEuler–Maclaurin summation formula and integral transforms such as theLaplace andMellin transforms. Repeatedintegration by parts will often lead to an asymptotic expansion.

Since aconvergentTaylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies anon-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known assuperasymptotics.^[2] The error is then typically of the form~ exp(−c/ε) whereε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such asBorel resummation to the divergent tail. Such methods are often referred to ashyperasymptotic approximations.

Seeasymptotic analysis andbig O notation for the notation used in this article.

First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion.

If${\displaystyle {\phi }_n}$  is a sequence ofcontinuous functions on some domain, and if${\displaystyle L}$  is alimit point of the domain, then the sequence constitutes anasymptotic scale if for everyn,

${\displaystyle {\phi }_{n+1}(x)=o({\phi }_n(x))(x\to L).}$ 

(${\displaystyle L}$  may be taken to be infinity.) In other words, a sequence of functions is an asymptotic scale if each function in the sequence grows strictly slower (in the limit${\displaystyle x\to L}$ ) than the preceding function.

If${\displaystyle f}$  is a continuous function on the domain of the asymptotic scale, thenf has an asymptotic expansion of order${\displaystyle N}$  with respect to the scale as a formal series

${\displaystyle \sum _{n=0}^N{a}_n{\phi }_n(x)}$ 

if

${\displaystyle f(x)-\sum _{n=0}^{N-1}{a}_n{\phi }_n(x)=O({\phi }_N(x))(x\to L)}$ 

or the weaker condition

${\displaystyle f(x)-\sum _{n=0}^{N-1}{a}_n{\phi }_n(x)=o({\phi }_{N-1}(x))(x\to L)}$ 

is satisfied. Here,${\displaystyle o}$  is thelittle o notation. If one or the other holds for all${\displaystyle N}$ , then we write^{[citation needed]}

${\displaystyle f(x)\sim \sum _{n=0}^{\mathrm{\infty }}{a}_n{\phi }_n(x)(x\to L).}$ 

In contrast to a convergent series for${\displaystyle f}$ , wherein the series converges for anyfixed${\displaystyle x}$  in the limit${\displaystyle N\to \mathrm{\infty }}$ , one can think of the asymptotic series as converging forfixed${\displaystyle N}$  in the limit${\displaystyle x\to L}$  (with${\displaystyle L}$  possibly infinite).

• Gamma function (Stirling's approximation)$${\displaystyle \frac{e^x}{x^x\sqrt{2\pi x}}\mathrm{\Gamma }(x+1)\sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots (x\to \mathrm{\infty })}$$ 
• Exponential integral$${\displaystyle x{e}^x{E}_1(x)\sim \sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^{n}n!}{x^n}(x\to \mathrm{\infty })}$$ 
• Logarithmic integral$${\displaystyle \mathrm{li}(x)\sim \frac{x}{\ln x}\sum _{k=0}^{\mathrm{\infty }}\frac{k!}{(\ln x{)}^k}}$$ 
• Riemann zeta function$${\displaystyle \zeta (s)\sim \sum _{n=1}^N{n}^{-s}+\frac{N^{1-s}}{s-1}-\frac{N^{-s}}{2}+N^{-s}\sum _{m=1}^{\mathrm{\infty }}\frac{B_{2m}{s}^{\overline{2m-1}}}{(2m)!N^{2m-1}}}$$ where${\displaystyle B_{2m}}$  areBernoulli numbers and${\displaystyle s^{\overline{2m-1}}}$  is arising factorial. This expansion is valid for all complexs and is often used to compute the zeta function by using a large enough value ofN, for instance${\displaystyle N>|s|}$ .
• Error function$${\displaystyle \sqrt{\pi }x{e}^{x^2}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x)\sim 1+\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{(2n-1)!!}{(2x^2{)}^n}(x\to \mathrm{\infty })}$$  where(2n − 1)!! is thedouble factorial.

Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of itsdomain of convergence. Thus, for example, one may start with the ordinary series

${\displaystyle \frac{1}{1-w}=\sum _{n=0}^{\mathrm{\infty }}{w}^n.}$ 

The expression on the left is valid on the entirecomplex plane${\displaystyle w\ne 1}$ , while the right hand side converges only for . Multiplying by${\displaystyle e^{-w/t}}$  and integrating both sides yields

${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{e^{-\frac{w}{t}}}{1-w}dw=\sum _{n=0}^{\mathrm{\infty }}{t}^{n+1}{\int }_0^{\mathrm{\infty }}{e}^{-u}{u}^{n}du,}$ 

after the substitution${\displaystyle u=w/t}$  on the right hand side. The integral on the left hand side, understood as aCauchy principal value, can be expressed in terms of theexponential integral. The integral on the right hand side may be recognized as thegamma function. Evaluating both, one obtains the asymptotic expansion

${\displaystyle e^{-\frac{1}{t}}\mathrm{Ei}\left(\frac{1}{t}\right)=\sum _{n=0}^{\mathrm{\infty }}n!t^{n+1}.}$ 

Here, the right hand side is clearly not convergent for any non-zero value oft. 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${\displaystyle \mathrm{Ei}\left(\frac{1}{t}\right)}$  for sufficiently smallt. Substituting${\displaystyle x=-\frac{1}{t}}$  and noting that${\displaystyle \mathrm{Ei}(x)=-E_1(-x)}$  results in the asymptotic expansion given earlier in this article.

Using integration by parts, we can obtain an explicit formula^[3]$${\displaystyle \mathrm{Ei}(z)=\frac{e^z}{z}\left(\sum _{k=0}^n\frac{k!}{z^k}+e_n(z)\right),e_n(z)\equiv (n+1)!z{e}^{-z}{\int }_{-\mathrm{\infty }}^z\frac{e^t}{t^{n+2}}dt}$$ For any fixed${\displaystyle z}$ , the absolute value of the error term${\displaystyle |e_n(z)|}$  decreases, then increases. The minimum occurs at${\displaystyle n\sim |z|}$ , at which point${\displaystyle |e_n(z)|\le \sqrt{\frac{2\pi }{|z|}}{e}^{-|z|}}$ . This bound is said to be "asymptotics beyond all orders".

For a given asymptotic scale${\displaystyle \{{\phi }_n(x)\}}$  the asymptotic expansion of function${\displaystyle f(x)}$  is unique.^[4] That is the coefficients${\displaystyle \{a_n\}}$  are uniquely determined in the following way: where${\displaystyle L}$  is the limit point of this asymptotic expansion (may be${\displaystyle \pm \mathrm{\infty }}$ ).

A given function${\displaystyle f(x)}$  may have many asymptotic expansions (each with a different asymptotic scale).^[4]

An asymptotic expansion may be an asymptotic expansion to more than one function.^[4]

• Asymptotic analysis
• Singular perturbation

• Watson's lemma
• Mellin transform
• Laplace's method
• Stationary phase approximation
• Method of dominant balance
• Method of steepest descent

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