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.

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 xa Laurent scale when xa Laurent scale when x±∞ Puiseaux scale when xa
• 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 \$Assumptions assumptions to make about parameters Direction Automatic direction in which x approaches a GenerateConditions Automatic whether to generate answers that involve conditions on parameters Method Automatic method to use PerformanceGoal \$PerformanceGoal aspects of performance to optimize
• Possible settings for Direction include:
•  Reals or "TwoSided" from both real directions "FromAbove" or -1 from above or larger values "FromBelow" or +1 from below or smaller values Complexes from 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:
•  Automatic nongeneric conditions only True all conditions False no conditions None return 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}:

 In:= Out= Find asymptotic approximations of solutions for x near 0:

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

 In:= Out= Find asymptotic approximations of solutions of a system of equations:

 In:= Out= Applications(12)

Introduced in 2019
(12.0)