Asymptotic expansion
Encyclopedia
In mathematics
an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré
) is a formal series of functions which has the property that truncating 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.
If φn 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,
. If f is a continuous function on the domain of the asymptotic scale, then f has an asymptotic expansion of order N with respect to the scale as a formal series 
if
or
If one or the other holds for all N, then we write
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 powers. 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.
Since a convergent Taylor series
fits the definition of asymptotic expansion as
well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. 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 as superasymptotics. The error is then typically of the form
where ε 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 as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
) is a formal series of functions which has the property that truncating 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.
If φn 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,
. If f is a continuous function on the domain of the asymptotic scale, then f has an asymptotic expansion of order N with respect to the scale as a formal series ifor
If one or the other holds for all N, then we write
See asymptotic analysis
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
and big O notation
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
for the notation.
The most common type of asymptotic expansion is a power series in either positive
or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and
integral transforms such as the Laplace and Mellin
Mellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
transforms. Repeated integration by parts will often lead to an asymptotic expansion.
Since a convergent Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
fits the definition of asymptotic expansion as
well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. 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 as superasymptotics. The error is then typically of the form
where ε 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 as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.Examples of asymptotic expansions
- Gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
-
- Exponential integralExponential integralIn mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.-Definitions:For real, nonzero values of x, the exponential integral Ei can be defined as...
-
>where
are Bernoulli numbers and 
is a rising factorial. This expansion is valid for all complex s and is often used to compute the zeta function by using a large enough value of N, for instance 
.
- Error functionError functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
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
The expression on the left is valid on the entire complex plane
, while the right hand side converges only for 
. Multiplying by 
and integrating both sides yields
after the substitution
on the right hand side.
The integral on the left hand side, understood as a Cauchy principal valueCauchy principal valueIn mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.-Formulation:...
, can be expressed
in terms of the exponential integralExponential integralIn mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.-Definitions:For real, nonzero values of x, the exponential integral Ei can be defined as...
. The integral on the right hand side may be recognized as the
gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. Evaluating both, one obtains the asymptotic expansion
Here, the right hand side is clearly not convergent for any non-zero value of t. 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
for sufficiently small t. Substituting 
and noting that 
results in the asymptotic expansion given earlier in this article.
- Error function
- Exponential integral
