Asymptotic

Warning: this article is under construction.

Asymptotic is statement about behaviors of a function (usually a holomorphic function) in some vicinity of some its singularity or at large values of the argument.

The main application of Asymptotic is approximation of the function, replacement of a function to another function, that is easier to evaluate, for those ranges of values of the argument, where the function and its asymptotic have similar properties (and similar values).

Asymptotic of a function \(f\) at a point \(C\) is a function \(A\) such that \[ \lim_{z\to C}\ (f(z)-A(z))\,B(z)=0, \] where \( |B(z)|\to\infty \) as \( z\to C \).

Here \(C\) may be a complex number or \( \infty \).

In many cases, additional restrictions on the values of \(z\) are imposed in this limit; these restrictions define the domain of validity of the asymptotic.

Function \(B\) qualifies, characterizes the precision of the asymptotic.

Often, \(B(z) \) is expressed as some power of \(z-C\) or (if \(C\) is zero or \(\infty\)) just some power function of \(z\).

Combinations of \(z\) with \(\log(z) \) or with \(\log(z-C) \) also can be used as \(B(z)\).

Conditional limit

Some restrictions 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.

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 asymptotic \(A\) is qualified as Strip asymptotic.

Sectorial asymptotic

The restriction may refer to the phase of the complex number \(z\); it allowed to have values from 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 finite number \(C\), the range of allowed \(z\) can be expressed with

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

with \( t_{\min}<t<t_{\max}\),

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.

More cases

The definitions above happen to be useful to classify asymptotics used for the evaluation (and the numerical implementation) of functions described in book «Superfunctions» .

In general, any restriction of values of \(z\) for the asymptotic can be specified and the corresponding asymptotic may get the specific name.

If some kind of restrictions is used many times, then it may have sense to define a new term to specify the corresponding asymptotic - in a similar way, as it is done in the subsections above.

However, the researcher is free to invent a new word, any Spunk, any «Кукарямба» and give this term some specific meaning required to simplify, to shorten the deduction or the description of some (perhaps also invented) objects. The hopes are allowed that the colleagues also find this term appropriate and begin to use it un the same or similar meaning(s).

Applications

Functions \(A\) and \(B\) in the Preamble are assumed to be much simpler to evaluate than the approximated function \(f\).

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

The asymptotic can be presented as a sum, series with indefinite number of terms.

Then, the asymptotic is called also Asymptotic series.

In general, the Asymptotic series diverge.
In some special cases (Taylor expansion, Laurent series), the Asymptotic series converge, but even in this case the radius of convergence is limited by the distance to the closest singularity of the holomorphic function \(f\); and even for an entire function \(f\) , the Taylor expansion is not a good representation for the large values of the argument.

For a non-trivial special function, several different asymptotics may be required to cover the complex plane with appropriate primary approximations. For example, this is realized in the C++ implementations fsexp.cin and fslog.cin of the natural tetration tet and the natural arctetration ate.

Acknowledgement

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References

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

Keywords

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

«Кукарямба», «Суперфункции»,