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Figure 1. Aysmptotic approximation: The function f(x)=x is an asymptotic approximation for the function f(x)=x+e^-x for large positive values of x.

In mathematical analysis, asymptotic analysis, also known as asymptotics, is the development and application of methods that generate an approximate analytical solution to a mathematical problem when a variable or parameter assumes a value that is large, small or near a specified value.[1]

An example of an asymptotic approximation is the function that accurately approximates the function for large positive values (Figure 1). For any desired accuracy, there is a corresponding range of values where this accuracy occurs. In this case, a chosen accuracy with a relative error of less than 1% occurs when the values are greater than 3.4.

Figure 2. Thomas Joannes Stieltjes: Stieltjes was a major contributor to the field of aysmptotic analysis.
Figure 3. Henri Poincaré: Poincaré was a major contributor to the field of aysmptotic analysis.

History[edit]

Henri Poincaré and Thomas Joannes Stieltjes independently developed the foundations of asymptotic analysis in 1886 (Figures 2-3).[2][3][4] Poincaré's focus was the "formal, analytic properties of those series" while Stieltjes's focus was to find "practical approximations for various functions and integrals."[5] Poincaré later applied this approach in his work on celestial mechanics, developing techniques of continuing importance.[2][6] Beginning in the early 20th century, asymptotic analysis became especially important in singular perturbation theory and the nonlinear equations of fluid mechanics.[6][7] Subsequent developments have led to applications in many areas of mathematics including computer science, analysis of algorithms, differential equations, integrals, functions, series, partial sums, and difference equations.[8][9][10]

Asymptotic relations[edit]

The continuous functions and of parameter or independent variable are defined on domain with element within the closure of .[11]

Big-O notation[edit]

The function is of order as approaches a finite number , written with big-O notation as , if there exists positive constant , independent of , and a neighborhood of meeting this condition. [12][13]

For approaching infinity, the big O-notation indicates there exists positive numbers and meeting this condition. [12]
The big-O notation may apply to all elements in a set .[14]
If is nonzero for near , except possibly at , then indicates that the quotient is bounded.[15]

Little-o notation[edit]

The function is much less than as approaches , written as , if for any positive number there is a neighborhood of meeting this condition.[12][13][15][16]

The relation is lower order than as approaches , written using little-o notation , is identical to the much less than relation .[12]

If is nonzero for near , except possibly at , then much less than indicates that the the quotient has limit 0 as approaches .[15]

Asymptotic equivalence[edit]

The function is equivalent to as approaches , written as , if this condition holds.[14]

If is nonzero for near , except possibly at , then indicates that the quotient has a limit 1 as approaches .[15]
For these asymptotic relations, the function is called the gauge function.[17]

Properties[edit]

The zero function, , can never be equivalent to any other function.[18]

The much less than () relation has the partial ordering property defined as if and then .[18]

Asymptotic equivalence has reflexive,symmetric and transitive properties. Additional properties are[19]

  • and a real number implies
  • and implies

Asymptotically equivalent functions remain asymptotically equivalent under integration if requirements related to convergence are met. There are more specific requirements for asymptotically equivalent functions to remain asymptotically equivalent under differentiation.[20]

Figure 4. Asymptotic integral approximation: The function f1(z) is an asymptotic accurate approximation of this integral for low z values.
Figure 5. Asymptotic integral approximation: The function f1(z) is an asymptotic accurate approximation of this integral for high z values.

Asymptotic theory[edit]

Asymptotic expansion[edit]

A sequence of functions, , defined on domain is an asymptotic sequence (scale) as approaches if each function is much less than (lower order) than the preceding function of the sequence, .[21]

Given an asymptotic sequence, , an asymptotic expansion (series) to terms of function is defined as this series.[22]

An asymptotic representation is a 1-term asymptotic sequence.[22] A truncated asymptotic expansion is an asymptotic expansion containing a finite number of terms. An asymptotic expansion of any number of terms, possibly infinite, has this form.[22]
The truncated asymptotic expansion is an asymptotic approximation to function as approaches if functions and are equivalent as approaches . An asymptotic approximation may not contain the optimum number of terms for the most accurate approximation when the variable is in a specified range.

Stieltjes integral[edit]

The Stieltjes integral's asymptotic expansion provides understanding of why an asymptotic series diverges and how to select the optimum number of terms for the asymptotic expansion; the general Stieltjes integral extends these insights to a broad range of asymptotic expansions (Table 1).[23][24]

Table 1. Stieltjes and general Stieltjes integrals
Stieltjes integral General Stieltjes integral
Function and integral
Moment integral
Asymptotic expansion
Remainder
Function, series, and remainder
Optimum number of terms ()
Remainder upper bound

Divergent asymptotic expansions[edit]

The Stieltjes asymptotic expansion arises from the Taylor series expansion of and converges for or equivalently for the finite range . However, the expansion coefficients arise from integration using this same variable as the integration variable over the semi-infinite range to , a range beyond where the Taylor series is valid. This leads to a divergent asymptotic expansion and the need to truncate the series after a finite number of terms. The limited range of convergence for the Taylor series used to construct the asymptotic series is a common cause for divergent asymptotic expansions.[25]

Beyond all orders feature[edit]

The approximation error for the Stieltjes asymptotic power series as an approximation for the Stieltjes integral contains a factor making the approximation error a non-analytic function. This means that to fully represent integrals like the Stieltjes integral may require exponential factors with the form not expressible as a power series. A factor of this type that may be unrecognized because it does not occur in a power series is described as a beyond all orders feature.[26]

Optimal expansion truncation[edit]

An N-term asymptotic series expansion optimally approximates a general Stieltjes integral if the error bound for the remainder for a N*-term expansion is less than the error bound for a (N*+1)-term expansion and less than or equal to the error bound for a (N*-1)-term expansion. This leads to the criteria for the optimum number of expansion terms (N*).

The general Stieltjes optimal truncation formula relies on finding the expansion's smallest term, , for a value of and truncating the asymptotic expansion just before the smallest term. This is the optimal truncation rule and commonly generalizes to other expansion types.[24] A similar optimal truncation rule is to truncate the expansion by excluding all terms greater than the smallest term.[27] Application of this formula to the Stieltjes asymptotic expansion with factorial coefficients leads to the formula .[23]

Superasymptotic approximation[edit]

A superasymptotic approximation is an optimally truncated asymptotic expansion. Superasymptotic approximations have an error on the order of with a positive constant and the optimum expansion contains a number of terms proportional to . As an example, the Stieltjes asymptotic expansion with factorial coefficients has an approximate error of and the optimum number of asymptotic expansion terms is the largest integer less than .[28]

Hyperasymptotic expansion[edit]

A hyperasymptotic approximation is an optimally truncated asymptotic approximation (superasymptotic approximation) with additional terms to represent the superasymptotic's remainder, the term that the superasymptotic expansion fails to represent. This may require different "scaling assumptions" and leads improved accuracy.[29]




 The general Stieltjes integral is an integral with this form.[24]

The Stieltjes moment integral is defined as[24]

.

The general Stieltjes series is defined as[24]

.


A more detailed analysis reveals that the part of the Stieltjes integral occurring over the range where the Taylor series is non-convergent is bounded by a term proportional to . This means that this part of the integral cannot be expressed as a power series expansion in because is a non-analytic function. Thus, this term will contribute to a minimum error, an error in computing the integral using any power series expansion in .[23]

The optimal truncation rule states that for any fixed value of the parameter or variable , truncating the asymptotic series just before the smallest term will give the most accurate approximation. This applies to Stieltjes series, but also commonly generalizes to other series types. [24]

Regularisation[edit]

Regularization is "the removal of the infinity in the remainder of a divergent series; regularised values can be evaluated for elementary series outside their circles of absolute convergence."[30] Instead of truncating a series and ignoring the terminal divergent part of the series, this terminal divergent series is assigned a regularised value, the terminant.[30] This approach identifies an integral that would generate this same divergent series, evaluates this integral and assigns this value to the terminant. This is feasible because the integral is assigned a finite value using methods like the residue theorem. One approach relies on Borel summation and a second approach relies on the Mellin inversion theorem (Mellin-Barnes regularisation).[31]

Generating asymptotic expansions[edit]

Asymptotic expansions from differential equations[edit]

For homogeneous linear differential equations, solutions may arise as Taylor series, and Frobenius series; asymptotic solutions may arise from dominant balance, phase integral (Wentzel–Kramers–Brillouin, Liouville–Green) and multiple-scale analysis methods.[32][33] Asymptotic series also arise as perturbation series solutions.[34] Using the Mellin transform, slowly converging series may be converted to accurate asymptotic series containing a small number of terms.[35]

Asymptotic expansions from integrals[edit]

Asymptotic expansions approximating integrals are generated by these methods:[36]

Asymptotic expansions from sums[edit]

The Euler–Maclaurin formula generates an asymptotic expansion approximating a sum.[36]

Summation of asymptotic expansions[edit]

There are methods that may accelerate the summation of slowly converging asymptotic expansions[36]

Converting a series to an integral[edit]

The sub-representation method may generate an integral representation from the function's asymptotic expansion. It may then be possible to use methods such as Laplace's method, stationary phase method or method of deepest descent to accurately evaluate this integral.[37]

The function's asymptotic expansion is known

.

From a table of function series, a function with similar terms, called the kernel is selected[38]

.

From another table, an appropriate sub-representation with functions and are selected that satisfies

.

The integral representation is by means of a h-transform[38][37]

.

Examples[edit]

Table 2. Asymptotic representations
Asymptotic representations
Prime-counting function
Factorial function
Partition function
Airy function
Hankel functions
The prime counting function counts the number of primes less than or equal to its argument.
The partition function is the number of ways (combinations) of writing a positive integer as a sum of positive integer addends.
The Hankel functions are solutions to the Bessel differential equation.
The Airy function is a solution to the differential equation with many applications in physics.
Table 3. Asymptotic expansions
Asymptotic expansions
Gamma function
Exponential integral
Error function
m!! is the double factorial
Table 4. Stieltjes series for an integral

Calculate the moment integral

Integral

Substitute and

Apply Stieltjes series formula

Substitute for and

Table 5. Asymptotic expansion approximates an integral

First approximate this integral for small z-values using a Taylor series.

.

The asymptotic expansion is

Due to the alternating sign of series terms, the approximation will be an average of a 3-term and 4-term series

Next approximate this integral for large z-values. Assign constants

with .

Integration by parts establishes this recurrence relation

Repeated application of the recurrence relation generates this asymptotic expansion

Due to the alternating sign of series terms, the approximation will be an average of a 1-term and 2-term series

The number of terms in each asymptotic series was arbitrary but comparison to the numerically integrated integral show the asymptotic expansions are accurate (Figures 4 and 5).

Applications[edit]

Differential equations[edit]

Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena.[39]

An illustrative example is the derivation of the boundary layer equations from the full Navier-Stokes equations governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter, ε: in the boundary layer case, this is the non-dimensional ratio of the boundary layer thickness to a typical length scale of the problem.[39] Applications of asymptotic analysis in mathematical modelling often center around a non-dimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.[39]

Statistics and probability theory[edit]

In mathematical statistics and probability theory, asymptotics are used in analysis of long-run or large-sample behavior of random variables and estimators.

Asymptotic theory provides limiting approximations of the probability distribution of sample statistics, such as the likelihood ratio statistic and the expected value of the deviance. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics. However, non-asymptotic bounds are provided by methods of approximation theory.

In mathematical statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables Zi for i = 1, …, n, for some positive integer n. An asymptotic distribution allows i to range without bound, that is, n is infinite.

A special case of an asymptotic distribution is when the late entries go to zero—that is, the Zi go to 0 as i goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.

This is based on the notion of an asymptotic function which cleanly approaches a constant value (the asymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.

The Edgeworth series provides an asymptotic approximations of probability distributions.

Geometry[edit]

An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation y becomes arbitrarily small in magnitude as x increases.

Applied mathematics[edit]

In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions.

Computer science[edit]

In computer science in the analysis of algorithms, considering the performance of algorithms.

Models of physical systems[edit]

Asymptotic analysis describes the behavior of physical systems, an example being statistical mechanics. Feynman graphs are an important tool in quantum field theory and the corresponding asymptotic expansions often do not converge.

Asymptotic analysis applies to accident analysis when identifying the causation of crash through count modeling with large number of crash counts in a given time and space.

Asymptotic versus Numerical Analysis[edit]

Debruijn illustrates the use of asymptotics in the following dialog between Miss N.A., a Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:[40]

N.A.: I want to evaluate my function for large values of , with a relative error of at most 1%.

A.A.: .

N.A.: I am sorry, I don't understand.

A.A.:

N.A.: But my value of is only 100.

A.A.: Why did you not say so? My evaluations give

N.A.: This is no news to me. I know already that .

A.A.: I can gain a little on some of my estimates. Now I find that

N.A.: I asked for 1%, not for 20%.

A.A.: It is almost the best thing I possibly can get. Why don't you take larger values of ?

N.A.: !!! I think it's better to ask my electronic computing machine.

Machine: f(100) = 0.01137 42259 34008 67153

A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.

N.A.: !!! . . .  !

Some days later, Miss N.A. wants to know the value of f(1000), but her machine would take a month of computation to give the answer. She returns to her Asymptotic Colleague, and gets a fully satisfactory reply.[40]

See also[edit]

Citations[edit]

  1. ^ Murray 2012, p. 1.
  2. ^ a b Poincaré 1886.
  3. ^ Stieltjes 1886.
  4. ^ Poincaré 1892.
  5. ^ Boven, Wesselink & Wepster} 2012.
  6. ^ a b Murray 2012, p. 2.
  7. ^ Verhulst 2006, p. 1.
  8. ^ Murray 2012.
  9. ^ Paulsen 2013.
  10. ^ Estrada & Kanwal 2012.
  11. ^ Bleistein & Handelsman 1986, p. 6,7.
  12. ^ a b c d Paulsen 2013, pp. 6, 7.
  13. ^ a b Estrada & Kanwal 2012, pp. 2, 3.
  14. ^ a b de Bruijn 1981, p. 4.
  15. ^ a b c d Bleistein & Handelsman 1986, pp. 6, 7.
  16. ^ Paulsen 2013, pp. 3.
  17. ^ Murray 2012, p. 3.
  18. ^ a b Paulsen 2013, pp. 1–3, 7.
  19. ^ Paulsen 2014, p. 9.
  20. ^ Olver 1974, pp. 8, 9, 21.
  21. ^ Erdelyi 1955, p. 8.
  22. ^ a b c Erdelyi 1955, p. 11-12.
  23. ^ a b c d Boyd 1999, pp. 13–17.
  24. ^ a b c d e f Bender & Orszag 2013, pp. 121–122.
  25. ^ Dingle 1972, p. 3.
  26. ^ Boyd 1999, pp. 7–8, 13–17.
  27. ^ Boyd 1999, p. 9.
  28. ^ Boyd 1999, p. 10.
  29. ^ Boyd 1999, p. 7.
  30. ^ a b Kowalenko 2011, p. 370.
  31. ^ Kowalenko 2011, pp. 388–404.
  32. ^ Bender & Orszag 1999.
  33. ^ White 2010, pp. 49–51.
  34. ^ Bender & Orszag 1999, pp. 331–428.
  35. ^ Dingle 1972, p. 26-55.
  36. ^ a b c Bender & Orszag 1999, pp. 247–302.
  37. ^ a b Dingle 1972, p. 56-99.
  38. ^ a b Bleistein et al.
  39. ^ a b c Howison 2005.
  40. ^ a b de Bruijn 1981, p. 19.

References[edit]

  • Balser, W. (1994), From Divergent Power Series To Analytic Functions, Springer-Verlag, ISBN 9783540485940

External links[edit]