Asymptotic Series

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Asymptotic Series

If φ_''n'' is a sequence of continuous functions on some domain, and if ''L'' is a (possibly infinite) limit point of the domain, then the sequence
constitutes an Asymptotic Scale if for every ''n'',

arphi_{n+1}(x) = o(arphi_n(x)) \  (x 
ightarrow L)

. If ''f'' is a continuous function on the domain of the asymptotic scale, then an asymptotic expansion of
''f'' with respect to the scale is a formal series

\sum_{n=0}^\infty a_n arphi_{n}(x)

such that, for any fixed ''N'',
:

f(x) = \sum_{n=0}^N a_n arphi_{n}(x) + O(arphi_{N+1}(x)) \  (x 
ightarrow L).

In this case, we write
:

f(x) \sim \sum_{n=0}^\infty a_n arphi_n(x)  \  (x 
ightarrow L)

.
See Asymptotic Analysis and Big O Notation for the notation.

The most common type of asymptotic expansion is a power series in either positive
or negative terms. While a convergent Taylor Series fits the definition as
given, a non-convergent series is what is usually intended by the phrase. Methods of generating such expansions include the Euler-Maclaurin Summation Formula and
integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion.

EXAMPLES OF ASYMPTOTIC EXPANSIONS

Gamma Function

::

rac{e^x}{x^x \sqrt{2\pi x}} \Gamma(x+1) \sim 1+rac{1}{12x}+rac{1}{288x^2}-rac{139}{51840x^3}-\cdots
 \  (x 
ightarrow \infty)

Exponential Integral

::

xe^xE_1(x) \sim \sum_{n=0}^\infty rac{(-1)^nn!}{x^n} \   (x 
ightarrow \infty)

Riemann Zeta Function

::

\zeta(s) \sim \sum_{n=1}^{N-1}n^{-s} + rac{N^{1-s}}{s-1} +
N^{-s} \sum_{m=1}^\infty rac{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 ''s'' and is often used to compute the zeta function by using

DETAILED 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

:

rac{1}{1-w}=\sum_{n=0}^\infty w^n

The expression on the left is valid on the entire complex plane

w

			
			

= \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^{-1/t}\; \operatorname{Ei}\left(rac{1}{t}
ight) = \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.

REFERENCES

Hardy, G. H., ''Divergent Series'', Oxford University Press, 1949

Paris, R. B. and Kaminsky, D., ''Asymptotics and Mellin-Barnes Integrals'', Cambridge University Press, 2001

Whittaker, E. and Watson, G. N., ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1963