AsymptoticRSolveValue
AsymptoticRSolveValue[eqn,f,x∞]
computes an asymptotic approximation to the difference equation eqn for f[x] near .
AsymptoticRSolveValue[{eqn_{1},eqn_{2},…},{f_{1},f_{2},…},x ∞]
computes an asymptotic approximation to a system of difference equations.
AsymptoticRSolveValue[eqn,f,x,ϵϵ_{0}]
computes an asymptotic approximation of f[x,ϵ] for the parameter ϵ centered at ϵ_{0}.
AsymptoticRSolveValue[eqn,f,…,{ξ,ξ_{0},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,…,xx_{0}] 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 x_{0} is g_{n}[x], then the result is AsymptoticLess[g[x]g_{n}[x],g_{n}[x]g_{n1}[x],xx_{0}] or g[x]g_{n}[x]∈o[g_{n}[x]g_{n1}[x]] as xx_{0}.
 The asymptotic approximation g_{n}[x] is often given as a sum g_{n}[x]α_{k}ϕ_{k}[x], where {ϕ_{1}[x],…,ϕ_{n}[x]} is an asymptotic scale ϕ_{1}[x]≻ϕ_{2}[x]≻⋯>ϕ_{n}[x] as xx_{0}. Then the result is AsymptoticLess[g[x]g_{n}[x],ϕ_{n}[x],xx_{0}] or g[x]g_{n}[x]∈o[ϕ_{n}[x]] as xx_{0}.
 Common asymptotic scales include:

Taylor scale when xx_{0} Laurent scale when xx_{0} Laurent scale when x±∞ Puiseaux scale when xx_{0}  The scales used to express the asymptotic approximation are automatically inferred from the problem and can often include more exotic scales.
 The center x_{0} 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:

AccuracyGoal Automatic digits of absolute accuracy sought Assumptions $Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate answers that involve conditions on parameters GeneratedParameters None how to name generated parameters Method Automatic method to use PerformanceGoal $PerformanceGoal aspects of performance to optimize PrecisionGoal Automatic digits of precision sought SeriesTermGoal Automatic number of terms in the approximation WorkingPrecision Automatic the 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
open allclose allBasic Examples (4)
Scope (11)
Basic Uses (2)
Ordinary Points (2)
Regular Singular Points (2)
Irregular Singular Points (3)
Asymptotic expansion for a linear firstorder OΔE with an irregular singular point at Infinity:
Asymptotic expansion for a linear secondorder OΔE with an irregular singular point:
Asymptotic expansions for a linear thirdorder OΔE with an irregular singular point:
Systems of ODEs (2)
Find a series solution for a linear system of two firstorder OΔEs at n=∞:
Plot the components of the solution:
Find a series solution for a linear system of three firstorder OΔEs at n=∞:
Compute the solution value for a particular choice of arbitrary constants and :
Find the value of the solution at using RecurrenceTable:
Applications (6)
Basic Applications (2)
Compute an approximate solution of a difference equation:
Compute a higherorder approximation:
Find an asymptotic approximation for Gamma:
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:
Obtain an approximate value for a member of the sequence:
Compare with the corresponding Fibonacci number:
Find an asymptotic approximation for the thirdorder 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:
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:
Obtain an approximate value for a member of the sequence:
Compare with the corresponding perturbed Fibonacci number:
Compute the leadingorder asymptotic term for the Apéry sequence, which satisfies the following linear secondorder difference equation:
Obtain the leading asymptotic term:
Verify that the first component of the expansion approaches 0 for large n:
Assign a value to based on the defining sum for the sequence:
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: