AsymptoticEquivalent

AsymptoticEquivalent[f,g,xx*]

gives conditions for as xx*.

AsymptoticEquivalent[f,g,{x1,,xn}{,,}]

gives conditions for as {x1,,xn}{,,}.

Details and Options

  • Asymptotic equivalent is also expressed as f is asymptotic to g and f is asymptotically equivalent to g. The point x* is often assumed from context.
  • Asymptotic equivalent is an equivalence relation and means TemplateBox[{{{f, (, x, )}, -, {g, (, x, )}}}, Abs]<=cTemplateBox[{{g, (, x, )}}, Abs] when x is near x* for all constants . It is a finer asymptotic equivalence relation than AsymptoticEqual.
  • Typical uses include simple expressions for functions and sequences near some point. It is frequently used for asymptotic solutions to equations.
  • For a finite limit point x* and {,,}:
  • AsymptoticEquivalent[f[x],g[x],xx*]for all there exists such that 0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<delta(c,x^*) implies TemplateBox[{{{f, (, x, )}, -, {g, (, x, )}}}, Abs]<=cTemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticEquivalent[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]for all there exists such that 0<TemplateBox[{{{, {{{x, _, 1}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<delta(epsilon,x^*) implies TemplateBox[{{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}, -, {g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}}, Abs]<=cTemplateBox[{{g, (, x, )}}, Abs]
  • For an infinite limit point:
  • AsymptoticEquivalent[f[x],g[x],x]for all there exists such that implies TemplateBox[{{{{f, (, x, )}, /, {g, (, x, )}}, -, 1}}, Abs]<=c
    AsymptoticEquivalent[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]for all there exists such that implies TemplateBox[{{{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}, /, {g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, -, 1}}, Abs]<=c
  • AsymptoticEquivalent[f[x],g[x],xx*] exists if and only if Limit[f[x]/g[x],xx*]1 when g[x] does not have an infinite set of zeros near x*.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    DirectionRealsdirection to approach the limit point
    GenerateConditionsAutomaticgenerate conditions for parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"what to optimize
  • Possible settings for Direction include:
  • Reals or "TwoSided"from both real directions
    "FromAbove" or -1from above or larger values
    "FromBelow" or +1from below or smaller values
    Complexesfrom all complex directions
    Exp[ θ]in the direction
    {dir1,,dirn}use direction diri for variable xi independently
  • DirectionExp[ θ] at x* indicates the direction tangent of a curve approaching the limit point x*.
  • Possible settings for GenerateConditions include:
  • Automaticnon-generic conditions only
    Trueall conditions
    Falseno conditions
    Nonereturn unevaluated if conditions are needed
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, Limit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (2)

Verify that as :

The ratio of the functions approaches as :

Verify that as :

The ratio of the functions approaches as and become large:

Scope  (10)

Compare functions that are not strictly positive:

Show that diverges at the same rate and with the same scale as at the origin:

Answers may be Boolean expressions rather than explicit True or False:

When comparing functions with parameters, conditions for the result may be generated:

By default, a two-sided comparison of the functions is made:

When comparing larger values of , vanishes at the same rate as x^2(1-cos(x)) TemplateBox[{{x}}, UnitStepSeq]:

The relationship fails for smaller values of :

Visualize the ratio of the two functions, showing it approaches 1 from above but not from below:

Functions like Sqrt may have the same relationship in both real directions along the negative reals:

If approached from above in the complex plane, the same relationship is observed:

However, approaching from below in the complex plane produces a different result:

This is due to a branch cut where the imaginary part of Sqrt reverses sign as the axis is crossed:

Hence, the relationship does not hold in the complex plane in general:

Visualize the quantity defining asymptotic equivalence approached from four principal complex directions:

Compare multivariate functions:

Visualize the norms of the two functions:

Compare multivariate functions at infinity:

Use parameters when comparing multivariate functions:

Options  (9)

Assumptions  (1)

Specify conditions on parameters using Assumptions:

Different assumptions can produce different results:

Direction  (5)

Equivalence from below:

Alternatively:

Equivalence from above:

Alternatively:

Equivalence at piecewise discontinuities:

Since it fails in one direction, the two-sided result is false as well:

Visualize the two functions and their ratio:

Equivalence at a pole is independent of the direction of approach:

Equality at a branch cut:

Determine equivalence, approaching from different quadrants:

Approaching the origin from the first quadrant:

Equivalently:

Approaching the origin from the second quadrant:

Approaching the origin from the right half-plane:

Approaching the origin from the bottom half-plane:

Visualize the ratio of the functions:

GenerateConditions  (3)

Return a result without stating conditions:

This result is only valid if n>0:

Return unevaluated if the results depend on the value of parameters:

By default, conditions are generated that return a unique result:

By default, conditions are not generated if only special values invalidate the result:

With GenerateConditions->True, even these non-generic conditions are reported:

Applications  (12)

Basic Applications  (5)

Show that two monomials equivalent at infinity have the same power and coefficient:

Use this to show that two polynomials that are equivalent have the same leading monomial:

Visualize three pairs of asymptotically equivalent polynomials:

Show that two monomials in equivalent at infinity have the same power and coefficient:

Use this to show that two polynomials in that are equivalent have the same leading monomial:

Visualize three pairs of asymptotically equivalent polynomials in :

Show that at 0, even though :

Visualize the functions:

Even though they are equal infinitely many times, consistently deviates from 1:

Show that at , even though :

Even though they are equal infinitely many times, consistently deviates from 1:

Show that at :

Asymptotic Approximation  (7)

A function approximates as with small relative error if as . Show that approximates with small relative error as :

Show that approximates with small relative error as :

But the preceding approximation does not have small absolute error:

Similarly, Stirling's formula approximation to has small relative error as :

But not small absolute error:

If is a function and an approximation to near , the approximation is asymptotic if at . In other words, the approximation has small relative error. Show that is an asymptotic approximation to at :

Show that is an asymptotic approximation to at :

However, is not an asymptotic approximation to at :

Stirling's formula gives an asymptotic approximation to as :

Show that is an asymptotic approximation to TemplateBox[{x}, PrimePi] as :

Another asymptotic approximation is given by TemplateBox[{x}, LogIntegral]:

Series generates an asymptotic approximation to elementary and special functions. For instance, generate a degree-10 approximation to at :

Show that the series is asymptotic:

Determine an asymptotic series of Cot[x] at 0:

Show that a series of Gamma[x] is asymptotic at -1:

Find an asymptotic approximation of at 0:

There can be subtleties with asymptotic approximation when the function to be approximated approaches zero infinitely many times in every neighborhood of the approximation point. As an example, consider the asymptotic expansion of TemplateBox[{1, x}, BesselJ] near :

The approximation cannot be shown to be asymptotic:

The problem is that at every zero of the Bessel function, the approximation is not quite zero:

Despite the ratio generally approaching one, TemplateBox[{{{J, (, {1, ,, x}, )}, -, besselJ}}, Abs]<=c TemplateBox[{{J, (, {1, ,, x}, )}}, Abs] is violated infinitely many times:

On the other hand, consider the approximation of the never-zero Hankel function TemplateBox[{1, x}, HankelH1]:

This approximation is asymptotic:

So is the approximation of the Hankel function of the second kind, TemplateBox[{1, x}, HankelH2]:

As TemplateBox[{1, x}, BesselJ]=1/2 (TemplateBox[{1, x}, HankelH1]+TemplateBox[{1, x}, HankelH2]), its approximation can be understood as nearly asymptotic, being the sum of two such approximations:

Alternatively, consider the asymptotic approximation of 1+TemplateBox[{1, x}, BesselJ] near :

This is a true asymptotic approximation:

The limit of the ratio of the approximation and the function approaches consistently:

Use AsymptoticIntegrate to generate asymptotic approximations to definite integrals. For instance, find an asymptotic approximation to as and compare to the exact value:

Create an asymptotic approximation with a smaller number of terms:

This approximation is asymptotic to the exact integral as well as the first approximation:

Use AsymptoticIntegrate to generate asymptotic approximations to indefinite integrals, though there is a need to account for the constant of integration. Consider an approximation of as :

Show that the approximation is asymptotic:

Compute two different asymptotic approximations of as :

There is no symbolic result to compare to, but the process can be shown to be asymptotic:

Use AsymptoticDSolveValue to generate asymptotic approximations to a differential equation:

There is no exact result to compare to, but the process can be shown to be asymptotic:

Compare with the values of a numerical solution obtained using NDSolveValue:

Properties & Relations  (4)

AsymptoticEquivalent is an equivalence relation, meaning it is reflexive ():

It is transitive ( and implies ):

And it is symmetric ( implies ):

AsymptoticEquivalent[f[x],g[x],xx0] iff Limit[f[x]/g[x],xx0]1:

In particular, if the limit is Indeterminate, :

If then :

The converse is false, so AsymptoticEquivalent is finer than AsymptoticEqual:

iff :

And similarly, iff :

Introduced in 2018
 (11.3)