User:TMM53/Asymptotic analysis

This page is about the behavior of functions as inputs approach infinity or some other limit value. For asymptotes in geometry, see Asymptote. For asymptotic distribution in statistics, see Asymptotic distribution.

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.

An example of an asymptotic approximation is the function ${\textstyle {\widetilde {y}}(x)=x}$ that accurately approximates the function ${\textstyle y(x)=x+e^{-x}}$ for large positive ${\textstyle x}$ values (Figure 1). For any desired accuracy, there is a corresponding range of ${\textstyle x}$ values where this accuracy occurs. In this case, a chosen accuracy with a relative error of less than 1% occurs when the ${\textstyle x}$ 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

Henri Poincaré and Thomas Joannes Stieltjes independently developed the foundations of asymptotic analysis in 1886 (Figures 2-3). 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." Poincaré later applied this approach in his work on celestial mechanics, developing techniques of continuing importance. Beginning in the early 20th century, asymptotic analysis became especially important in singular perturbation theory and the nonlinear equations of fluid mechanics. 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.

Asymptotic relations

Asymptotic notations

The parameter or independent variable ${\textstyle z}$ is a real or complex number. The continuous functions ${\textstyle f(z),g(z),h(z)}$ and $k(z)$ are defined on domain D with ${\textstyle z_{0}}$ contained in the closure of D. The neighborhood ${\textstyle N(\delta ,z_{0})}$ is contained in D, and is a pointed neighborhood because it excludes point ${\textstyle z_{0}}$; for example, ${\textstyle 0<|z-z_{0}|<\delta }$ is a pointed neighborhood.

The function ${\textstyle f(z)}$ is of order ${\textstyle g(z)}$ as ${\textstyle z}$ approaches a finite number ${\textstyle z_{0}}$, written with big-O notation as ${\textstyle f(z)=O(g(z))\ (z\to z_{0})}$, if there exists positive constant ${\textstyle M}$ and pointed neighborhood ${\textstyle N(\delta ,z_{0})}$, independent of ${\textstyle z}$, such that

$|f(z)|\leq M|g(z)|$ for ${\textstyle z}$ in $N(\delta ,z_{0})$.

For ${\textstyle z}$ approaching an infinite number,this means there exists positive numbers ${\textstyle M}$ and ${\textstyle L}$ such that

$|f(z)|\leq M|g(z)|$ for $z>L$.

The big-O notation may apply to all elements ${\textstyle z}$ in a set ${\textstyle S}$

$f(z)=O(g(z))\ (z\in S)$.

If ${\textstyle g(z)}$ is nonzero for ${\textstyle z}$ near ${\textstyle z_{0}}$, except possibly at ${\textstyle z_{0}}$, then ${\textstyle f(z)=O(g(z))\ (z\to z_{0})}$ indicates that the quotient ${\textstyle |f(z)/g(z)|}$ is bounded.

The function ${\textstyle f(z)}$ is much less than ${\textstyle g(z)}$ as ${\textstyle z}$ approaches ${\textstyle z_{0}}$, written as ${\textstyle f(z)\ll g(z)\ (z\to z_{0})}$, if for any ${\textstyle c>0}$ there is a pointed neighborhood ${\textstyle N(\delta ,z_{0})}$ such that

$|f(z)|\leq c|g(z)|$ for $z$ in ${\textstyle N(\delta ,z_{0})}$.

The relation ${\textstyle f(z)}$ is lower order than ${\textstyle g(z)}$ as ${\textstyle z}$ approaches ${\textstyle z_{0}}$, written using little-o notation ${\textstyle f(z)=o(g(z))\ (z\to z_{0})}$, is identical to the relation ${\textstyle f(z)\ll g(z)\ (z\to z_{0})}$.

If ${\textstyle g(z)}$ is nonzero for ${\textstyle z}$ near ${\textstyle z_{0}}$, except possibly at ${\textstyle z_{0}}$, then ${\textstyle f(z)\ll g(z)\ (z\to z_{0})}$ indicates that the limit of the quotient ${\textstyle |f(z)/g(z)|}$ is 0 as ${\textstyle z}$ approaches ${\textstyle z_{0}}$

$\operatorname {Lim} \ {\frac {f(z)}{g(z)}}=0\ {\text{as}}\ z\to z_{0}$.

The function ${\textstyle f(z)}$ is equivalent to ${\textstyle g(z)}$ as ${\textstyle z}$ approaches ${\textstyle z_{0}}$, written as ${\textstyle f(z)\sim g(z)\ (z\to z_{0})}$, if this condition holds $f(z)=g(z)(1+o(1))\ z\to z_{0}$.

If ${\textstyle g(z)}$ is nonzero for ${\textstyle z}$ near ${\textstyle z_{0}}$, except possibly at ${\textstyle z_{0}}$, then ${\textstyle f(z)\sim g(z)\ (z\to z_{0})}$ indicates that the limit of the quotient ${\textstyle |f(z)/g(z)|}$ is 1 as ${\textstyle z}$ approaches ${\textstyle z_{0}}$

$\operatorname {Lim} \ {\frac {f(z)}{g(z)}}=1\ {\text{as}}\ z\to z_{0}$.

For these asymptotic relations, the function ${\textstyle g(z)}$ is called the gauge function.

Properties

The zero function, ${\textstyle f(z)=0}$, can never be equivalent to any other function.

The much less than (${\textstyle \ll }$) relation has the partial ordering property defined as if ${\textstyle f(z)\ll g(z)}$ and ${\textstyle g(z)\ll h(z)}$ then ${\textstyle f(z)\ll h(z)}$.

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

• $f(z)\sim g(z)\ (z\to z_{0}\ )$ and $\ r\$ a real number implies $(f(z))^{r}\sim (g(z))^{r}\ (z\to z_{0})$
• $f(z)\sim g(z)\ (z\to z_{0})\$ and $\ h(z)\sim k(z)\ (z\to z_{0})\$ implies ${\frac {f(z)}{h(z)}}\sim {\frac {g(z)}{k(z)}}\ (z\to z_{0})$

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.

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 expansions

Main article: Asymptotic expansion

An asymptotic expansion (series) is a series that may accurately approximate a functions over a region of the function's domain; for example, when the function's argument is small, large or neighbors a point (Figures 4,5). Asymptotic series include many types of series including a function's Taylor series. Asymptotic series may more accurately approximate a function than the function's Taylor series with an equal number of terms. An asymptotic series may also approximate non-analytic functions that lack a Taylor series such as functions with an essential singularity. Unlike a Taylor series, some asymptotic series are not necessarily convergent and the most accurate asymptotic series may be a truncated series with a finite number of terms.

Definitions

A sequence of functions, ${\textstyle \{g_{0}(z),g_{1}(z),\ldots \}}$ defined on domain ${\textstyle D}$ is an asymptotic sequence (scale) as ${\textstyle z}$ approaches ${\textstyle z_{0}}$ if for each pair of consecutive functions, ${\textstyle g_{m+1}(z)=o(g_{m}(z))\ (z\to z_{0})}$.

Given an asymptotic sequence, ${\textstyle \{g_{0}(z),g_{1}(z),\ldots \}}$, an asymptotic expansion (series) to ${\textstyle N}$ terms of function ${\textstyle f(z)}$ is defined as this series

$f(z)=\sum _{m=0}^{N}a_{m}g_{m}(z)+o(g_{N}(z))\ (z\to z_{0})$.

An asymptotic representation is a 1-term asymptotic sequence.

An asymptotic expansion of any number of terms, possibly infinite, is written as

$f(z)\sim \sum _{m=0}a_{m}g_{m}(z)\ (z\to z_{0})$.

Truncation

The general Stieltjes integral provides understanding on the optimum number of terms to sum in a asymptotic expansion. The Stieltjes integral is an integral with this form:

$f(z)=\int _{0}^{\infty }{\frac {\rho (t)}{(1+zt)}}\ dt$.

The Stieltjes moment integral is defined as

$a_{n}=\int _{0}^{\infty }t^{n}\rho (t)\ dt$.

The general Stieltjes series is defined as

$f(z)\sim \sum _{n=0}^{\infty }\ a_{n}(-z)^{n}\quad (z\to 0+)$.

The Stieltjes series arises from the Taylor series expansion of ${\textstyle (1+zt)^{-1}}$ that converges for ${\textstyle |zt|<1}$. However, the moment integral relies on ${\textstyle t}$ extending to infinity. So for very large ${\textstyle t}$, ${\textstyle z}$ must be near 0 to keep ${\textstyle |zt|<1}$. 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 series used to construct the asymptotic series is a common cause for divergent asymptotic expansions.

The optimal truncation rule states that for any fixed value of the parameter or variable ${\textstyle z}$, 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.

Generating asymptotic expansions

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. Using the Mellin transform, slowly converging series may be converted to accurate asymptotic series containing a small number of terms.

Asymptotic expansions approximating integrals are generated by these methods:

• Taylor series
• Integration by parts
• Laplace's method
• Watson's lemma
• Stationary phase approximation
• Method of steepest descent

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

Perturbation methods generate asymptotic expansions for differential equations and integrals.

Summation of asymptotic expansions

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

• Shanks transformation
• Richardson extrapolation
• Euler summation
• Borel summation
• Padé approximant

Converting a series to an integral

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

The function's series representation is known

$f(z)=\sum _{n=0}^{\infty }\ a_{n}z^{n}$.

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

$k(z)=\sum _{n=0}^{\infty }\ b_{n}z^{n}$.

From another table, an appropriate sub-representation with functions ${\textstyle g(t)}$ and ${\textstyle h(t)}$ are selected that satisfies

${\frac {a_{n}}{b_{n}}}=\int \ g^{n}(t)\ h(t)\ dt$.

The integral representation is by means of a h-transform

$f(z)=\int \ k(zg(t))\ h(t)\ dt$.

Examples

Asymptotic representations

	Asymptotic representations
Prime-counting function	$\pi (z)\sim {\frac {z}{\ln z}}\quad (z\to +\infty )$
Factorial function	$$z!\sim {\sqrt {2\pi n}}\left({\frac {z}{e}}\right)^{z}\quad (z\to +\infty )$$
Partition function	$p(z)\sim {\frac {1}{4z{\sqrt {3}}}}e^{\pi {\sqrt {\frac {2z}{3}}}}\quad (z\to +\infty )$
Airy function	$\operatorname {Ai} (z)\sim {\frac {e^{-{\frac {2}{3}}z^{\frac {3}{2}}}}{2{\sqrt {\pi }}z^{1/4}}},\ {\text{for }}-\pi <\arg z<\pi \quad (|z|\to +\infty )$
Hankel functions	$${\begin{aligned}H_{\alpha }^{(1)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-\pi <\arg z<2\pi ,\quad (|z|\to +\infty )\\H_{\alpha }^{(2)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-2\pi <\arg z<\pi \quad (|z|\to +\infty )\\\end{aligned}}$$
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 ${\textstyle y^{\prime }-xy=0}$ with many applications in physics.

Asymptotic expansions

	Asymptotic expansions
Gamma function	$${\frac {e^{z}}{z^{z}{\sqrt {2\pi z}}}}\Gamma (z+1)\sim 1+{\frac {1}{12z}}+{\frac {1}{288z^{2}}}-{\frac {139}{51840x^{3}}}-\cdots \ (z\to \infty )$$
Exponential integral	$$ze^{z}E_{1}(z)\sim \sum _{n=0}^{\infty }{\frac {(-1)^{n}n!}{z^{n}}}\ (z\to \infty )$$
Error function	$${\sqrt {\pi }}ze^{z^{2}}\operatorname {erfc} (z)\sim 1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {(2n-1)!!}{n!(2z^{2})^{n}}}\ (z\to \infty )$$ m!! is the double factorial

Stieltjes series for an integral

Calculate the moment integral $a_{n}=\int _{0}^{\infty }\ t^{n}e^{-t}\ dt$ $a_{n}=n!$ Integral $f(s)=\int _{0}^{\infty }\ {\frac {e^{-w/s}}{1-w}}\ dw$ Substitute $z=-s$ and $t=w/s$ $f(-z)=(-z)\int _{0}^{\infty }\ {\frac {e^{-t}}{1+tz}}\ dt$ Apply Stieltjes series formula $f(-z)\sim (-z)\sum _{n=0}^{\infty }\ a_{n}(-z)^{n}$ Substitute for $a_{n}$ and $s=-z$ $f(s)\sim \sum _{n=0}^{\infty }\ n!\ s^{n+1}$

Asymptotic expansion approximates an integral

First approximate this integral for low z-values using a Taylor series. ${\begin{aligned}&{\sqrt {\frac {2}{\pi }}}\int _{z}^{\infty }e^{-t^{2}/2}\ dt\\&={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }e^{-t^{2}/2}\ dt-{\sqrt {\frac {2}{\pi }}}\int _{0}^{z}e^{-t^{2}/2}\ dt\\&=1-{\sqrt {\frac {2}{\pi }}}\int _{0}^{z}\left(1-{\frac {t^{2}}{2}}+{\frac {t^{4}}{8}}-{\frac {t^{6}}{48}}+\dots \right)\ dt\\&=1-{\sqrt {\frac {2}{\pi }}}\left(z-{\frac {z^{3}}{6}}+{\frac {z^{5}}{40}}-{\frac {z^{7}}{336}}+\ldots \right)\\\end{aligned}}$. The asymptotic expansion is ${\sqrt {\frac {2}{\pi }}}\int _{z}^{\infty }e^{-t^{2}/2}\ dt\sim 1-{\sqrt {\frac {2}{\pi }}}\left(z-{\frac {z^{3}}{6}}+{\frac {z^{5}}{40}}-{\frac {z^{7}}{336}}+\ldots \right)\ (z\to 0)$ Due to the alternating sign of series terms, the approximation will be an average of a 3-term and 4-term series $f_{1}(z)=1-{\sqrt {\frac {2}{\pi }}}\left(z-{\frac {z^{3}}{6}}+{\frac {z^{5}}{40}}-{\frac {z^{7}}{672}}\right)$ Next approximate this integral for high z-values. Assign constants $a_{0}=1,a_{1}=1,a_{2}=3\cdot 1,a_{3}=5\cdot 3\cdot 1,\dots$ with $a_{p+1}=(2p+1)a_{p}$. Integration by parts establishes this recurrence relation ${\begin{aligned}&a_{p}{\sqrt {\frac {2}{\pi }}}\int _{z}^{\infty }{\frac {e^{-t^{2}/2}}{t^{2p}}}\ dt\\&=-a_{p}{\sqrt {\frac {2}{\pi }}}\int _{z}^{\infty }{\frac {d(e^{-t^{2}/2})}{t^{2p+1}}}\ dt\\&=\left[-a_{p}{\sqrt {\frac {2}{\pi }}}{\frac {e^{-t^{2}/2}}{t^{2p+1}}}\right]_{z}^{\infty }-a_{p+1}{\sqrt {\frac {2}{\pi }}}\int _{z}^{\infty }{\frac {e^{-t^{2}/2}}{t^{2(p+1)}}}\ dt\\&=a_{p}{\sqrt {\frac {2}{\pi }}}{\frac {e^{-z^{2}/2}}{z^{2p+1}}}-a_{p+1}{\sqrt {\frac {2}{\pi }}}\int _{z}^{\infty }{\frac {e^{-t^{2}/2}}{t^{2(p+1)}}}\ dt\end{aligned}}$ Repeated application of the recurrence relation generates this asymptotic expansion ${\sqrt {\frac {2}{\pi }}}\int _{z}^{\infty }e^{-t^{2}/2}\ dt\sim {\sqrt {\frac {2}{\pi }}}e^{-z^{2}/2}\left({\frac {1}{z}}-{\frac {1}{z^{3}}}+{\frac {3\cdot 1}{z^{5}}}-{\frac {5\cdot 3\cdot 1}{z^{7}}}\ldots \right)\ (z\to \infty )$ Due to the alternating sign of series terms, the approximation will be an average of 1-term and 2-term series $f_{1}(z)={\sqrt {\frac {2}{\pi }}}e^{-z^{2}/2}\left({\frac {1}{z}}-{\frac {1}{2z^{3}}}\right)$ These asymptotic expansions accurately approximate the integral for low and high values of the argument ${\textstyle z}$ (Figures 4,5).

Applications

Asymptotic analysis is used in several mathematical sciences.

Differential equations

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

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. 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.

Statistics and probability theory

Main article: Asymptotic distribution

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 Z_i 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 Z_i 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

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 $y={\frac {1}{x}},$ y becomes arbitrarily small in magnitude as x increases.

Applied mathematics

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

Computer science

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

Models of physical systems

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

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

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

A.A.: $f(x)=x^{-1}+\mathrm {O} (x^{-2})\qquad (x\to \infty )$.

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

A.A.: $|f(x)-x^{-1}|<8x^{-2}\qquad (x>10^{4}).$

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

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

$|f(x)-x^{-1}|<57000x^{-2}\qquad (x>100).$

N.A.: This is no news to me. I know already that $0<f(100)<1$.

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

$|f(x)-x^{-1}|<20x^{-2}\qquad (x>100).$

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 $x$?

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.

See also

• Asymptote – Limit of the tangent line at a point that tends to infinity
• Asymptotic computational complexity – in theory of computationPages displaying wikidata descriptions as a fallback
• Asymptotic density – Concept in number theoryPages displaying short descriptions of redirect targets
• Asymptotic theory (statistics) – Study of convergence properties of statistical estimators
• Asymptotology – Dealing with applied mathematical systems in limiting cases
• Big O notation – Describes limiting behavior of a function
• Leading-order term – Terms in a mathematical expression with the largest order of magnitude
• Method of dominant balance – Solution of a simplified form of an equation
• Method of matched asymptotic expansions
• Watson's lemma – lemma on the asymptotic behavior of integralsPages displaying wikidata descriptions as a fallback

Citations

References

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

• Bender, Carl M.; Orszag, Steven A. (2013). Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer Science & Business Media. ISBN 978-1-4757-3069-2.

• Bleistein, Norman; Handelsman, Richard A. (1986). Asymptotic Expansions of Integrals. Courier Corporation. ISBN 978-0-486-65082-1.

• Breitung, Karl W. (2006). Asymptotic Approximations for Probability Integrals. Springer. ISBN 978-3-540-49033-3.

• de Bruijn, N. G. (1981), Asymptotic Methods in Analysis, Dover Publications, ISBN 9780486642215

• Estrada, R.; Kanwal, R. P. (2002), A Distributional Approach to Asymptotics, Birkhäuser, ISBN 9780817681302

• Estrada, Ricardo; Kanwal, Ram P. (2012). Asymptotic Analysis: A Distributional Approach. Springer Science & Business Media. ISBN 978-1-4684-0029-8.

• Howison, Sam (2005). Practical Applied Mathematics: Modelling, Analysis, Approximation. Cambridge University Press. ISBN 978-0-521-84274-7.

• Miller, P. D. (2006), Applied Asymptotic Analysis, American Mathematical Society, ISBN 9780821840788

• Murray, J. D. (1984), Asymptotic Analysis, Springer, ISBN 9781461211228

• Murray, J. D. (2012). Asymptotic Analysis. Springer Science & Business Media. ISBN 978-1-4612-1122-8.

• Olver, F. W. J. (2014). Introduction to Asymptotics and Special Functions. Academic Press. ISBN 978-1-4832-6708-1.

• Paris, R. B.; Kaminsky, D. (2001), Asymptotics and Mellin-Barnes Integrals, Cambridge University Press

• Paulsen, William (2013). Asymptotic Analysis and Perturbation Theory. CRC Press. ISBN 978-1-4665-1512-3.

• Poincaré, Henri (1892). "Chapitre VII. Solutions asymptotiques". Les méthodes nouvelles de la mécanique céleste Tome I (PDF). Paris: Gauthier-Villars. pp. 335–382.

• Poincaré, Henri (1886). "Sur les intégrales irrégulières: Des équations linéaires" (PDF). Acta Math. 8: 295–344.

• Stieltjes, T.J. (1886). "Recherches sur quelques series semi-convergentes" (PDF). Annales Scientifiques de l'École Normale Supérieure, Ser. 3, Band 3: 201–258.

• van Boven, Hasse; Wesselink, Rob; Wepster, Steven (2012). "Asymptotic series of Poincaré and Stieltjes" (PDF). Nieuw archief voor wiskunde. 5. 13.3: 187–190.

• Verhulst, Ferdinand (2006). Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics. Springer Science & Business Media. ISBN 978-0-387-28313-5.

• Yasutaka, Sibuya (1975). Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient. Elsevier. ISBN 978-0-08-087129-5.

External links

• Asymptotic Analysis  —home page of the journal, which is published by IOS Press
• A paper on time series analysis using asymptotic distribution