Asymptotic expansion

The asymptotic expansion of the function f (x) is a formal functional series such that the sum of an arbitrary finite number of members of this series approximates ( approximates ) the function f (x) in a neighborhood of some (possibly infinitely distant) limit point . The concept of an asymptotic expansion of a function and an asymptotic series was introduced by Henri Poincare in solving problems of celestial mechanics . Separate cases of asymptotic decomposition were discovered and applied as early as the 18th century. Asymptotic expansions and series play an important role in various problems of mathematics , mechanics, and physics .

Content

Definition

Let functions $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ satisfy the property: $\varphi _{n+1}(x)=o(\varphi _{n}(x))\ (x\rightarrow L)\quad \forall n\in \mathbb {N}$ ${\ displaystyle \ varphi _ {n + 1} (x) = o (\ varphi _ {n} (x)) \ (x \ rightarrow L) \ quad \ forall n \ in \ mathbb {N}}$ for some limit point $L$ ${\ displaystyle L}$ domain of function f (x) . Function sequence $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ satisfying the indicated conditions is called an asymptotic sequence. Row: $\sum _{n=0}^{\infty }a_{n}\varphi _{n}(x)$ ${\ displaystyle \ sum _ {n = 0} ^ {\ infty} a_ {n} \ varphi _ {n} (x)}$ for which the conditions are satisfied: $f(x)-\sum _{n=0}^{N-1}a_{n}\varphi _{n}(x)=O(\varphi _{N}(x))\ (x\rightarrow L)$ ${\ displaystyle f (x) - \ sum _ {n = 0} ^ {N-1} a_ {n} \ varphi _ {n} (x) = O (\ varphi _ {N} (x)) \ ( x \ rightarrow L)}$

or equivalently:

f(x)-\sum _{n=0}^{N-1}a_{n}\varphi _{n}(x)=o(\varphi _{N-1}(x))\ (x\rightarrow L).

{\ displaystyle f (x) - \ sum _ {n = 0} ^ {N-1} a_ {n} \ varphi _ {n} (x) = o (\ varphi _ {N-1} (x)) \ (x \ rightarrow L).}

is called the asymptotic expansion of the function f (x) or its asymptotic series. This fact is reflected:

f(x)\sim \sum _{n=0}^{\infty }a_{n}\varphi _{n}(x)\ (x\rightarrow L).

{\ displaystyle f (x) \ sim \ sum _ {n = 0} ^ {\ infty} a_ {n} \ varphi _ {n} (x) \ (x \ rightarrow L).}

The difference between a convergent series and an asymptotic expansion for a function $f(x)$ ${\ displaystyle f (x)}$ can be illustrated as follows: for a convergent series for any fixed $x$ ${\ displaystyle x}$ the series converges in value $f(x)$ ${\ displaystyle f (x)}$ at $N\rightarrow \infty$ ${\ displaystyle N \ rightarrow \ infty}$ , while in the asymptotic expansion for a fixed $N$ ${\ displaystyle N}$ the series converges in value $f(x)$ ${\ displaystyle f (x)}$ in the limit $x\rightarrow L$ ${\ displaystyle x \ rightarrow L}$ ( $L$ ${\ displaystyle L}$ may be infinite).

Asymptotic Erdeia decomposition

The asymptotic Erdeyi decomposition has a more general definition. Row $\sum _{n=0}^{\infty }a_{n}\varphi _{n}(x)$ ${\ displaystyle \ sum _ {n = 0} ^ {\ infty} a_ {n} \ varphi _ {n} (x)}$ is called the asymptotic Erdeyi decomposition of f (x) if there exists such an asymptotic sequence $\psi _{n}$ ${\ displaystyle \ psi _ {n}}$ , what

f(x)-\sum _{n=0}^{N}a_{n}\varphi _{n}(x)=o(\psi _{N}(x))\ (x\rightarrow L).

{\ displaystyle f (x) - \ sum _ {n = 0} ^ {N} a_ {n} \ varphi _ {n} (x) = o (\ psi _ {N} (x)) \ (x \ rightarrow L).}

This fact is written as follows:

f(x)\sim \sum _{n=0}^{\infty }a_{n}\varphi _{n}(x)\ (x\rightarrow L)\quad \{\psi _{n}(x)\}.

{\ displaystyle f (x) \ sim \ sum _ {n = 0} ^ {\ infty} a_ {n} \ varphi _ {n} (x) \ (x \ rightarrow L) \ quad \ {\ psi _ { n} (x) \}.}

Such a generalized expansion has many common properties with the usual asymptotic expansion, however, the theory of such a decomposition is poorly studied, often little useful for numerical calculations, and rarely used.

Examples

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\rightarrow \infty )

{\ displaystyle {\ 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 \ rightarrow \ infty)}

Integral exponential function

xe^{x}E_{1}(x)\sim \sum _{n=0}^{\infty }{\frac {(-1)^{n}n!}{x^{n}}}\ (x\rightarrow \infty )

{\ displaystyle xe ^ {x} E_ {1} (x) \ sim \ sum _ {n = 0} ^ {\ infty} {\ frac {(-1) ^ {n} n!} {x ^ {n }}} \ (x \ rightarrow \ infty)}

Riemann Zeta Function

\zeta (s)\sim \sum _{n=1}^{N-1}n^{-s}+{\frac {N^{1-s}}{s-1}}+N^{-s}\sum _{m=1}^{\infty }{\frac {B_{2m}s^{\overline {2m-1}}}{(2m)!N^{2m-1}}}

{\ displaystyle \ zeta (s) \ sim \ sum _ {n = 1} ^ {N-1} n ^ {- s} + {\ frac {N ^ {1-s}} {s-1}} + N ^ {- s} \ sum _ {m = 1} ^ {\ infty} {\ frac {B_ {2m} s ^ {\ overline {2m-1}}} {(2m)! N ^ {2m-1 }}}}

Where

B_{2m}

{\ displaystyle B_ {2m}}

- Bernoulli numbers and

s^{\overline {2m-1}}=s(s+1)(s+2)\cdots (s+2m-2)

{\ displaystyle s ^ {\ overline {2m-1}} = s (s + 1) (s + 2) \ cdots (s + 2m-2)}

. This decomposition is valid for all complex s .

Error function

{\sqrt {\pi }}xe^{x^{2}}{\rm {erfc}}(x)\sim 1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {(2n)!}{n!(2x)^{2n}}}.

{\ displaystyle {\ sqrt {\ pi}} xe ^ {x ^ {2}} {\ rm {erfc}} (x) \ sim 1+ \ sum _ {n = 1} ^ {\ infty} (- 1 ) ^ {n} {\ frac {(2n)!} {n! (2x) ^ {2n}}}.}

An example of the asymptotic expansion of Erdeyi, which is not an ordinary decomposition, is ^[1] :

{\frac {\sin(x)}{x}}\sim \sum _{n=0}^{\infty }{\frac {n!e^{-(n+1)x/2n}}{(\log x)^{n}}}\quad (x\rightarrow \infty )\ \{(\log x)^{-n}\}.

{\ displaystyle {\ frac {\ sin (x)} {x}} \ sim \ sum _ {n = 0} ^ {\ infty} {\ frac {n! e ^ {- (n + 1) x / 2n }} {(\ log x) ^ {n}}} \ quad (x \ rightarrow \ infty) \ \ {(\ log x) ^ {- n} \}.}

Notes

↑ Roderick Wong. Asymptotic approximations of integrals. Academic Press, London, 1989 13

Literature

Mathematical Encyclopedia / Ed. I.M. Vinogradova. Volume 2 - M.: Mir, 1985.
Erdeyi A. Asymptotic expansions / Per. from English - M., 1962
Bleistein, N. and Handlesman, R., Asymptotic Expansions of Integrals, Dover, New York, 1975

[1] Roderick Wong. Asymptotic approximations of integrals. Academic Press, London, 1989 13