Asymptotic

Warning: this article is under construction.

Asymptotic \(A\) of function \(f\) at point \(z_0\) on the domain \(D\in\mathbb C\) is special kind of approximation, characterized in that, that its quality improves while the input of function, remaining within domain \(D\) approaches \(z_0\). Here, \(z_0\) is either a complex number or \(\infty\).

Asymptotics are qualified by the criteria of the "quality" of the approximation, by the kind of the limiting point \(z_0\) and by the shape of domain \(D\).

Usually, asymptotic \(A\) is easier to evaluate than approximated function \(f\).

Sometimes, for some applications, the properties of asymptotic coincide with those of the approximated function. In these cases, following the last 6th of the TORI axioms, the concept that uses asymptotic, has priority.

Asymptotics can be used for fast, qualitative estimates, but also for the precise evaluation of the approximated function.

Apology

There are several various definitions of term «Asymptotic» at various sites, t various dictionaries. Often, term «Asymtotic» is used in an euistic mode, assuming, that «Everybody nows» and with hope, that the professional pure mathematicians understand the problem and redo it with mathematical rugosity.
Often, sigh a hope is not supported with interest and pacience of so-called «pure mathematicians; one example of such a case is resented in the fairy tale «Lesson of quantum topology».
According to legend, the Greatest genius of all centuries and all civilizations gensek Mao Zedong used to say, that one may count only with own efforts . At the testing of Chinese nuclear weapon, the people were not even alerted. Even worse, it was assumed, that at the use, the significant part of population of China will be dead, and the gensek still will have sufficient number of slaves to maintain his luxury style of life. No any opposition of this idea had been allowed in 1964's China: every citizen could count only whit his own efforts..

Following the Ideas of the Greatest genius of all centuries and all civilizations, this article suggests definition of term Asymptotic in the next section.

Definition

Assume function \(f\) is defined at some domain \(B \in \mathbb C\)

Assume, some point \(z_0\) is specified; it can be a comely number or infinity.

Assume some domain \(D\in B\) is specified.

Assume some special function \(A\) is defined at \(D\).

Assume some holomorphic function \(\tau\) is defined a the set of values of function \(f\).

Let \[ \lim_{z\to z_0, \ z\in D} \Big( \tau\big(f(z)\big) - \tau\big(A(z)\big) \Big) =0 \]

Then, function \(A\) is asymtoric of function \(f\) at domain \(D\) with the criterion function \(\tau\).

The same can be written as follows: \[ f(z) \underset{\mathrm{\tau, \ z\in D,\ z\to z_0}}{\sim} A(z) \]

If function \(\tau\) is not specified, by default, it is assumed to be identity function.

Function \(\ R(z)=f(z)\!-\!A(z) \ \) is called Residual of asymtotic \(A\) of function \(f\)

Function \[\ a(z)=-\lg\left(\frac {|f(z)\!-\!A(z)|} {|f(z)|\!+\!|A(z)|} \right) \]

is called agreement of asymptotic \(A\) with function \(f\).

The agreement gives the criterion of quality of the asymptotic \(A\) at the use as the primary approximation for the numerical implementation of function \(f\).

For the approximation of a function, it is desirable that residual approach zero, and the agreement remain big, say, comparable to the amount of decimal digits that are supported in arithmetic used in the numerical implementation. The most of numerical implementations loaded at TORI provide of order of 14 significant figures.

For some asymptotics, it is not possible to achieve \(R\to 0\).

Then, as criterion, the "small residual" \[ r(z)=\tau(f(z))\!-\!\tau(A(z)) \] can be used; function \[ \alpha (z)= -\lg\left(\frac {|\tau(f(z)) - \tau(A(z))|} {|\tau(f(z))|+ |\tau(A(z))|} \right) \] can be used instead of the agreement; it is desirable to provide small values of \(|r(z)|\) and keep big values of \(\alpha(z)\) at least for \(z\) in vicinity of \(z_0\).

Notes about the definition above

  \(u\!+\!\mathrm i v\!=\!\mathrm{Filog}(x\!+\mathrm i y)\), bottom of Fig.18.2
in book «Superfunctions»

Function \(A\) is supposed to be already known and well described. There is no sense to approximate some unknown function with anther unknown function. Ideally, asymtorotic \(A\) is elementary function, but, generally, it can be any Special function.

In actions, talking about asumtotics, often one does not specify the domain \(D\), assuming, that it is obvious. Usually, it is imlicitly assumed, that \(\tau\) is either identity function or natural logarithm.

In general, some function may have several asymtoirics with different domains \(D\) - especially is \(z_0=\infty\) or \(z_0\) belongs to the branch cut of function \(f\).

One example of such a function is Filog; its complex map is shown in figure at right.
Filog\((\beta)\) expresses the fixed point of logarithm to base \(b=\mathrm e^\beta\) that is

\(\log_b = (z \!\mapsto\! \log(\beta z))\).

At real \(z_0 < \exp(1/\mathrm d)\), the asymptotic of function Filog depends on the way he input approaches this value.

If we extend the definition of asymptotic and allow the asymptotic to be a function of two arguments, then the tetration to base \(b=\exp(1/\mathrm e)\) can be considered as asymptotic of tetration to base \(b=\exp(\beta)\) while real part of \(\beta\) remains bigger than \(1/\mathrm e\); tetration being considered as function of logarithm of the base is not holomorphic at this point.

Conditional limit

Some restrictios can be applied to values of \(z\) that are allowed in the limit above.

If the imaginary part of \(z\) is assumed to remain in the interval \(a,b\), then, the asymptotic \(A\) is qualified as strip asymptotic.

"Fast" growth of \(B\) at \(C\) means that the asymptotic \(A\) is "good", robust, and is useful for the numeric implementation of function \(f\).

Two special cases of restrictions on values of \(z\) in the limit above are considered in the two subsections below.

Strip asymptotic

For the case of infinite \(C\), it may be assumed that the imaginary part of \(z\) is limited, \(a<\Im(z)<b\) where \(a\) and \(b\) are real number, and it is assumed that \(\Re(z) \to \infty\) or \(\Re(z) \to -\infty\) or \(\Re(z) \to +\infty\).

Then the asympotic \(A\) is qualified as Strip asymptotic.

Sectorial asymptotic

The restriction may refer to the phase of the compass number \(z\); it allowed to have values fro some sector at the complex plane:

\( z = r \ \exp(\mathrm i t) \)

where \(r\) is positive real number and \(t\) is real number from interval \( (t_{\mathrm{min}},t_{\mathrm{max}})\)

Then, the asymtiric \(A\) is qualified as Sectorial asymptotic.

For ignite number \(C\), the range of allowed \(z\) can be expressed with

\( z-C = r \ \exp(\mathrm i t) \)

The angle interval \((t_{\min},t_{\max})\) is fixed and does not depend on \(r\).

in this case, the asymptotic refers to the approach to value \(C\) from some direction, and it is also Sectorial asymptotic.

Examples

Gallery

Warning

The definition of term «Asymptotic» in TORI is «outstanding», it does not coincide with definition of the term at other dictionaries, cites, manuals, textbooks.

Often, the definition or term «asymptotic» assumes that function \(A\) is obtained by truncation of the series with positive or negative power of \(z-z_0\) in the formulas above. Here, no such an assumption is involved.

Editor tires to make the definition of term «asymptotic» so close to the popular one as possible, while avoiding contradictions.

The definition is a draft; it can be modified in order to fill better the examples of various asymptotics considered.

References

https://en.wikipedia.org/wiki/Asymptotic_analysis In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

https://api.pageplace.de/preview/DT0400.9781439864548_A38306604/preview-9781439864548_A38306604.pdf FrankW.J.Olver. Asymptotics and Special Functions Copyright O 1997by A K Peters, Ltd.

https://www.researchgate.net/publication/39064661_Asymptotics_and_Mellin-Barnes_Integrals Asymptotics and Mellin–Barnes Integrals September 2001 DOI:10.1017/CBO9780511546662 Authors: Richard Bruce Paris, David Kaminski// © Cambridge University Press 2001

https://dlmf.nist.gov/5.11 NIST Digital Library of Mathematical Functions (2026) 5 Gamma Function // Properties // §5.11(i) Poincaré-Type Expansions

Keywords

«Agreement», «Approximation», «Asymptotic», «Domain of validity», «Entire Function with Logarithmic Asymptotic», «Holomorphic function», «Numerical implementation», «Residual», «Sectorial asymptotic», «Special function», «Spunk», «Stirling», «Strip asymptotic», «Superfunctions»,

«Кукарямба», «Суперфункции»,