AsymptoticRSolveValue

AsymptoticRSolveValue[eqn,f,x]

computes an asymptotic approximation to the difference equation eqn for f[x] near .

AsymptoticRSolveValue[{eqn1,eqn2,},{f1,f2,},x ]

computes an asymptotic approximation to a system of difference equations.

AsymptoticRSolveValue[eqn,f,x,ϵϵ0]

computes an asymptotic approximation of f[x,ϵ] for the parameter ϵ centered at ϵ0.

AsymptoticRSolveValue[eqn,f,,{ξ,ξ0,n}]

computes the asymptotic approximation to order n.

Details and Options

  • Asymptotic approximations to difference equations are also known as asymptotic expansions, perturbation solutions, regular perturbations, etc. They are also known by specific methods used to compute some of them, such as Taylor series and Frobenius series.
  • Asymptotic approximations are typically used to solve problems for which no exact solution can be found or to get simpler answers for computation, comparison and interpretation.
  • AsymptoticRSolveValue[eqn,,xx0] computes the leading term in an asymptotic expansion for eqn. Use SeriesTermGoal to specify more terms.
  • If the exact result is g[x] and the asymptotic approximation of order n at x0 is gn[x], then the result is AsymptoticLess[g[x]-gn[x],gn[x]-gn-1[x],xx0] or g[x]-gn[x]o[gn[x]-gn-1[x]] as xx0.
  • The asymptotic approximation gn[x] is often given as a sum gn[x]αkϕk[x], where {ϕ1[x],,ϕn[x]} is an asymptotic scale ϕ1[x]ϕ2[x]>ϕn[x] as xx0. Then the result is AsymptoticLess[g[x]-gn[x],ϕn[x],xx0] or g[x]-gn[x]o[ϕn[x]] as xx0.
  • Common asymptotic scales include:
  • Taylor scale when xx0
    Laurent scale when xx0
    Laurent scale when x±
    Puiseaux scale when xx0
  • The scales used to express the asymptotic approximation are automatically inferred from the problem and can often include more exotic scales.
  • The center x0 can be any finite or infinite real or complex number.
  • The order n must be a positive integer and specifies order of approximation for the asymptotic solution. It is not related to polynomial degree.
  • The following options can be given:
  • AccuracyGoalAutomaticdigits of absolute accuracy sought
    Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate answers that involve conditions on parameters
    GeneratedParametersNonehow to name generated parameters
    MethodAutomaticmethod to use
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoalAutomaticdigits of precision sought
    SeriesTermGoalAutomaticnumber of terms in the approximation
    WorkingPrecisionAutomaticthe precision used in internal computations
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, AsymptoticRSolveValue typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (4)

Compute an asymptotic approximation for a difference equation:

Find a series solution for a difference equation:

Plot the solution:

Find an asymptotic expansion for a perturbation problem:

Plot the solution:

Find an asymptotic expansion for a functional equation:

Scope  (11)

Basic Uses  (2)

Compute a series solution of order 2 for an ordinary difference equation (OΔE):

Obtain series approximations with different numbers of terms:

Ordinary Points  (2)

Find a series solution for a linear first-order OΔE with an ordinary point at Infinity:

Plot the solution:

Series solution for a linear second-order OΔE with an ordinary point at Infinity:

Plot the solution:

Regular Singular Points  (2)

Frobenius series solution for a linear first-order OΔE with a regular singular point at Infinity:

Plot the solution:

Series solution for a linear second-order OΔE with a regular singular point at Infinity:

Plot the solution:

Irregular Singular Points  (3)

Asymptotic expansion for a linear first-order OΔE with an irregular singular point at Infinity:

Asymptotic expansion for a linear second-order OΔE with an irregular singular point:

Asymptotic expansions for a linear third-order OΔE with an irregular singular point:

Plot the solution using a logarithmic scale:

Systems of ODEs  (2)

Find a series solution for a linear system of two first-order OΔEs at n=:

Plot the components of the solution:

Find a series solution for a linear system of three first-order OΔEs at n=:

Compute the solution value for a particular choice of arbitrary constants and :

Find the value of the solution at using RecurrenceTable:

Compare with the approximate value:

Options  (1)

GeneratedParameters  (1)

Use differently named constants:

Use subscripted constants:

Applications  (6)

Basic Applications  (2)

Compute an approximate solution of a difference equation:

Compute a higher-order approximation:

Find an asymptotic approximation for Gamma:

Obtain an approximate value for a member of the sequence:

Compare with the exact value:

Special Sequences  (4)

Find an asymptotic approximation for the Fibonacci sequence, starting with the expansion for the difference equation satisfied by this sequence:

Verify that the first component of the expansion approaches 0 for large n:

Set the first component to 0:

Assign a value to :

Obtain an approximate value for a member of the sequence:

Compare with the corresponding Fibonacci number:

Find an asymptotic approximation for the third-order Fibonacci sequence, starting with the expansion for the difference equation satisfied by this sequence:

Verify that the second and third components of the expansion approach 0 for large n:

Set the first and third components to 0:

Assign a value to :

Obtain an approximate value for a member of the sequence:

Compare with the corresponding Fibonacci number:

Find an asymptotic approximation for the perturbed Fibonacci sequence, starting with the expansion for the difference equation satisfied by this sequence:

Verify that the first component of the expansion approaches 0 for large n:

Set the first component to 0:

Assign a value to :

Obtain an approximate value for a member of the sequence:

Compare with the corresponding perturbed Fibonacci number:

Compute the leading-order asymptotic term for the Apéry sequence, which satisfies the following linear second-order difference equation:

Obtain the leading asymptotic term:

Verify that the first component of the expansion approaches 0 for large n:

Set the first component to 0:

Assign a value to based on the defining sum for the sequence:

Obtain an approximate value for a member of the sequence:

Compare with the corresponding Apéry number:

Properties & Relations  (3)

Solutions satisfy the difference equation up to a given order:

Use RSolveValue to find an exact solution:

Use RecurrenceTable to find a numerical solution:

Introduced in 2019
 (12.0)
 |
Updated in 2020
 (12.1)