AsymptoticIntegrate

AsymptoticIntegrate[f,x,{x,x0,n}]

computes an asymptotic approximation of the indefinite integral for x centered at x0 of order n.

AsymptoticIntegrate[f,{x,a,b},{α,α0,n}]]

computes an asymptotic approximation of the definite integral for α centered at α0 of order n.

Details and Options

  • Asymptotic approximations to integrals are also known as asymptotic expansions and perturbation expansions. They are also known by specific methods to compute some of them, such as Laplace's method, method of stationary phase and method of steepest descent, etc.
  • 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.
  • If the exact result is g[x] and the asymptotic approximation of order n at x0 is gn[x], then 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 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:
  • Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsFalsewhether to generate answers that involve conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"aspects of performance to optimize
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, AsymptoticIntegrate typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (2)

Compute an asymptotic expansion for an integral:

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Compare with an expansion of the exact result:

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Obtain the leading term in the expansion for a Gaussian integral:

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Compute the required term:

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Compare with the result from a numerical approximation:

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Scope  (21)

Applications  (6)

Properties & Relations  (3)

Possible Issues  (1)

See Also

Integrate  AsymptoticLess  Series  NIntegrate  AsymptoticDSolveValue

Introduced in 2018
(11.3)