AsymptoticEqual

AsymptoticEqual[f,g,xx*]

gives conditions for or as xx*.

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

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

Details and Options

  • Asymptotic equal is also expressed as f is big-theta of g, f is bounded by g, f is of order g, and f grows as g. The point x* is often assumed from context.
  • Asymptotic equal is an equivalence relation and means c_1 TemplateBox[{{g, (, x, )}}, Abs]<=TemplateBox[{{f, (, x, )}}, Abs]<=c_2 TemplateBox[{{g, (, x, )}}, Abs] when x is near x* for some constants and . It is a coarser asymptotic equivalence relation than AsymptoticEquivalent.
  • Typical uses include expressing simple bounds for functions and sequences near some point. It is frequently used for asymptotic solutions to equations and to give simple lower bounds for computational complexity.
  • For a finite limit point x* and {,,}, the result is:
  • AsymptoticEqual[f[x],g[x],xx*]there exist , and such that 0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<delta(c_1,c_2,x^*) implies c_1 TemplateBox[{{g, (, x, )}}, Abs]<=TemplateBox[{{f, (, x, )}}, Abs]<=c_2 TemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticEqual[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]there exist , and such that 0<TemplateBox[{{{, {{{x, _, 1}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<delta(epsilon,x^*) implies c_1 TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=TemplateBox[{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=c_2 TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]
  • For an infinite limit point, the result is:
  • AsymptoticEqual[f[x],g[x],x]there exist , and such that implies c_1 TemplateBox[{{g, (, x, )}}, Abs]<=TemplateBox[{{f, (, x, )}}, Abs]<=c_2 TemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticEqual[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]there exist , and such that implies c_1 TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=TemplateBox[{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=c_2 TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]
  • AsymptoticEqual[f[x],g[x],xx*] exists if and only if MinLimit[Abs[f[x]/g[x]],xx*]>0 and MaxLimit[Abs[f[x]/g[x]],xx*]< 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:
  • Automaticnongeneric 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 :

In[1]:=
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Out[1]=

The former can be "sandwiched" between two constant multiples of the latter:

In[3]:=
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Out[3]=

Verify that as :

In[1]:=
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Out[1]=

The former can be "sandwiched" between two constant multiples of the latter:

In[2]:=
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Out[2]=

Scope  (9)

Options  (9)

Applications  (10)

Properties & Relations  (6)

See Also

AsymptoticEquivalent  AsymptoticLessEqual  AsymptoticGreaterEqual  AsymptoticLess  AsymptoticGreater  MaxLimit  MinLimit  AsymptoticIntegrate

Introduced in 2018
(11.3)