AsymptoticSolve

AsymptoticSolve[eqn,{y,b},{x,a,n}]

computes asymptotic approximations of solutions y[x] of the equation eqn passing through {a,b} of order n.

AsymptoticSolve[eqn,{y},{x,a,n}]

computes asymptotic approximations of solutions y[x] of the equation eqn for x near a.

AsymptoticSolve[eqns,{{y1,y2,},{b1,b2,}},{{x1,x2,},{a1,a2,},n}]

computes asymptotic approximations of solutions {y1[x1,x2,],y2[x1,x2,],} of the system of equations eqns.

AsymptoticSolve[,Reals]

computes only solutions that are real valued for real argument values.

Details and Options

  • 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.
  • The asymptotic approximation yn[x] is often given as a sum yn[x]αkϕk[x], where {ϕ1[x],,ϕn[x]} is an asymptotic scale ϕ1[x]ϕ2[x]>ϕn[x] as xa. Then the result satisfies AsymptoticLess[y[x]-yn[x],ϕn[x],xa] or y[x]-yn[x]o[ϕn[x]] as xa.
  • Common asymptotic scales include:
  • Taylor scale when xa
    Laurent scale when xa
    Laurent scale when x±
    Puiseaux scale when xa
  • The scales used to express the asymptotic approximation are automatically inferred from the problem and can often include more exotic scales.
  • The center coordinates a and b can be any finite or infinite real or complex numbers.
  • The order n must be a positive integer and specifies order of approximation for the asymptotic solution. It may not be related to polynomial degree.
  • The system of equations eqns can be any logical combination of equations.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions to make about parameters
    DirectionAutomaticdirection in which x approaches a
    GenerateConditionsAutomaticwhether to generate answers that involve conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal$PerformanceGoalaspects of performance 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*.
  • For finite values of a, the Automatic setting means from above.
  • When domain Reals is specified, the solutions are real valued when x approaches a in the indicated Direction.
  • 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, AsymptoticSolve typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (4)

Find asymptotic approximations of solutions passing through the point {0,0}:

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Find asymptotic approximations of solutions for x near 0:

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Find only the solutions that are real valued when x approaches 0 from above:

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Find asymptotic approximations of solutions of a system of equations:

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Scope  (18)

Options  (8)

Applications  (12)

Introduced in 2019
(12.0)