AsymptoticSolve

AsymptoticSolve[eqn,yb,x->a]

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

AsymptoticSolve[eqn,{y},x->a]

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,}]

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

AsymptoticSolve[eqns,,{{x1,x2,},{a1,a2,},n}]

computes the asymptotic approximation to order n.

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.
  • AsymptoticSolve[eqn,,xa] computes the leading term in an asymptotic expansion for eqn. Use SeriesTermGoal to specify more terms.
  • 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
    SeriesTermGoalAutomaticnumber of terms in the approximation
  • 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  (5)

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

Find asymptotic approximations of solutions for x near 0:

Find the leading terms of asymptotic approximations of solutions as :

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

Find asymptotic approximations of solutions of a system of equations:

Scope  (18)

One-Dimensional Solutions in 2D  (8)

Power series solutions of polynomial equations passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Power series solutions of analytic equations passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Puiseux series solutions of polynomial equations passing through a specified point:

Solutions that are real valued when x approaches 0 from above:

Solutions that are real valued when x approaches 0 from below:

Puiseux series solutions of analytic equations passing through a specified point:

Asymptotic series solutions passing through a specified point:

Solutions of polynomial equations near a specified value of the independent variable:

Real asymptotic series solutions at infinity:

Plot the asymptotic solution and the solution it approximates:

Equations with symbolic parameters:

Conditions on parameters may be generated:

One-Dimensional Solutions in nD  (5)

Power series solutions of polynomial systems passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Power series solutions of analytic systems passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Puiseux series solutions of polynomial systems passing through a specified point:

None of the solutions are real valued when t approaches 0 from above:

Two of the solutions are real valued when t approaches 0 from below:

Solutions of polynomial systems near a specified value of the independent variable:

Equations with symbolic parameters:

Higher-Dimensional Solutions in nD  (5)

Power series solutions of polynomial equations passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Power series solutions of analytic equations passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Power series solutions of polynomial systems passing through a specified point:

Power series solutions of analytic systems passing through a specified point:

Power series solutions of polynomial systems near specified values of independent variables:

Options  (9)

Assumptions  (1)

Specify conditions on parameters using Assumptions:

Different assumptions can produce different results:

Direction  (3)

By default, AsymptoticSolve gives solutions valid when x approaches 0 from above:

This finds the solutions valid when x approaches 0 from below:

Complex solutions may also depend on the direction:

This gives solutions that are real when x approaches 0 from a complex direction:

GenerateConditions  (3)

By default, AsymptoticSolve gives conditions it assumed to obtain the result:

This gives the result without the assumed conditions:

By default, assumed conditions that are generically true are not reported:

With GenerateConditions->True, all conditions are reported:

With GenerateConditions->None, AsymptoticSolve returns only generically valid results:

If nongeneric conditions are needed, AsymptoticSolve returns unevaluated:

Method  (1)

Return a series whenever the result is a power series or a Puiseux series:

Check that the series solutions satisfy the equation:

SeriesTermGoal  (1)

By default, AsymptoticSolve[eqn,,xa] computes the leading terms of the solutions:

Use SeriesTermGoal to obtain more terms:

Applications  (12)

Implicit Functions  (3)

The equation implicitly defines two different functions near each . Compute third-order asymptotic approximations for these two functions near :

Define functions based on these expansions:

At the point , these two functions have different values:

However, both exactly satisfy the equation at :

Visualize the equation and the approximations to its two branches:

In this case, it is easy to solve exactly for the two implicitly defined functions:

The two expressions returned by AsymptoticSolve are the series of the exact solutions:

Compute the second-order asymptotic approximations to the unit circle at :

The approximations closely track the circle of both larger and smaller values of at these regular points:

At the singular point , the approximation uses fractional powers:

Visually, the approximations seem to only be defined for values of :

This is because at the functions switch from being real to purely imaginary:

Trying to find solutions over the reals will therefore fail:

However, it is possible to find purely real expressions if restricting to smaller values of :

The curve crosses the line infinitely many times. On any section that passes the vertical line testany vertical line intersects the curve only once; no vertical line intersects the section more than oncea function is implicitly defined:

Compute an approximation for the section that goes through the origin:

Note that this matches the Taylor series of , the inverse function of :

Compute an approximation for the section that goes through the point :

Visualize the curve and the two approximations:

Perturbed Equations  (2)

Find solutions of a perturbed polynomial equation:

Plot the asymptotic solutions and the solutions they approximate:

Investigate the behavior of solutions of an analytic equation under a small perturbation:

Plot the asymptotic solutions and the solutions they approximate:

Series Solutions of Equations  (2)

Find a series solution of an equation at a nonsingular point:

The derivative of with respect to does not vanish at :

Find the series solution up to order five:

The result satisfies the equation:

Find a multivariate series solution of a system of equations at a nonsingular point:

The Jacobian of with respect to does not vanish at :

Find the series solution up to order three:

The result satisfies the equations:

Asymptotic Approximations of Curves  (3)

Find Puiseux series solutions in a neighborhood of a singular point of an algebraic plane curve:

Plot the asymptotic solutions and the curve they approximate near 0:

Find Puiseux series solutions in a neighborhood of a singular point of an algebraic space curve:

Plot the asymptotic solutions:

Find numeric solutions using asymptotic solution values as starting points:

Compare the numeric solutions and the asymptotic solutions:

Show the curve as an intersection of two surfaces:

Approximate Fermat's spiral near 0:

Compare plots:

Asymptotic Solutions of Physics Problems  (2)

Solve Kepler's equation for the eccentric anomaly in terms of the mean anomaly :

Compare with the exact solution for eccentricity :

Study the energy levels of a particle of mass in a one-dimensional box of width and depth . Solutions , and of the time-independent Schrödinger equation to the left of the box, inside the box, and to the right of the box are given by:

The solution must be continuously differentiable on the boundary of the box:

The homogenous linear equations admit nonzero solutions if their coefficient matrix is singular:

Assume that and are 1 and the units are chosen so that :

Find the possible energy levels for :

Compute the asymptotic solution near :

Compare the asymptotic solution and the minimum exact solution:

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