AsymptoticProduct

AsymptoticProduct[f,x,xx0]

computes an asymptotic approximation of the indefinite product for x near x0.

AsymptoticProduct[f,{x,a,b},αα0]

computes an asymptotic approximation of the definite product for α near α0.

AsymptoticProduct[f,,{ξ,ξ0,n}]

computes the asymptotic approximation to order n.

Details and Options

  • AsymptoticProduct is typically used to compute products for which no exact result can be found or to get simpler answers for computation, comparison and interpretation. In such cases, an asymptotic approximation often gives enough information for simplifying or solving application problems.
  • AsymptoticProduct[f,,ξξ0] computes the leading term in an asymptotic expansion for the product of f. Use SeriesTermGoal to specify more terms.
  • 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.
  • 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 α0 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 expansion. It is not related to polynomial degree.
  • The following options can be given:
  • AccuracyGoalAutomaticdigits of absolute accuracy sought
    Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate answers that involve conditions on parameters
    GeneratedParametersNonehow to name generated parameters
    MethodAutomaticmethod to use
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoalAutomaticdigits of precision sought
    RegularizationNonewhat regularization scheme to use
    SeriesTermGoalAutomaticnumber of terms in the approximation
    WorkingPrecisionAutomaticthe precision used in internal computations
  • With the default setting of Automatic for GenerateConditions, conditions on parameters are typically not returned in the results from AsymptoticProduct. Answers that include conditions on parameters may be obtained by setting GenerateConditions to True.
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, AsymptoticProduct typically solves more problems or produces simpler results, but it potentially uses more time and memory.
  • With the default setting of Automatic for WorkingPrecision, AccuracyGoal and PrecisionGoal, AsymptoticProduct may return an asymptotic approximation with a lower precision, even if the input has infinite precision.

Examples

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Basic Examples  (2)

Compute an asymptotic approximation for a product:

Compare with the exact value:

Improve the approximation using SeriesTermGoal:

Compute an asymptotic expansion for a product with respect to a parameter:

Compute the required expansion:

Compare with the exact value for small a:

Scope  (14)

Basic Uses  (3)

Compute an asymptotic approximation for an indefinite product:

Compute an asymptotic approximation for a definite product:

Compute an asymptotic approximation for a parametric product:

Indefinite Products  (5)

Compute an asymptotic expansion for a polynomial product:

Compare with the result given by Product:

Asymptotic expansion for the indefinite products of rational functions:

Asymptotic expansion for the indefinite product of a hypergeometric term:

Estimate the value at a point:

Compare with the value given by NProduct:

Asymptotic expansion for the indefinite product of a rational exponential function:

Asymptotic expansion for the indefinite product of an algebraic function:

Definite Products  (4)

Compute an asymptotic expansion for a rational product:

Compare with the result given by NProduct:

Asymptotic approximation for a rational-exponential product:

Compare with the result given by NProduct:

Compute an asymptotic approximation for a hypergeometric product:

Compare with the exact result:

Asymptotic expansion for an algebraic product:

Compare with the exact result:

Parametric Products  (2)

Asymptotic expansion for a finite product with respect to the parameter a:

Compute a numerical approximation:

Compare with the result given by NProduct:

Asymptotic expansion for an infinite product with respect to the parameter z:

Compute a numerical approximation:

Compare with the result given by NProduct:

Improve the asymptotic approximation by computing more terms:

Applications  (4)

Compute an approximation for a finite product:

Compute a numerical approximation for increasing values of n:

Compare with the exact results given by Product:

Compare with the series expansion of the exact symbolic formula:

Compute the value of an infinite rational product using an asymptotic expansion of the corresponding finite product:

Obtain the value of the infinite product:

Use DiscreteLimit to obtain the same answer:

Confirm the answer using Product:

Establish the convergence of an infinite product by computing the asymptotic expansion of the corresponding indefinite product:

Compute the exact value using an asymptotic expansion of the finite product:

Obtain the same result using Product:

Compute an asymptotic approximation for :

Compute the limiting value of the asymptotic expression:

Visualize the convergence to the limiting value:

Properties & Relations  (4)

AsymptoticProduct computes the product up to a given order:

Use Product to compute the product in closed form:

Use NProduct to compute a numerical approximation:

Compute an asymptotic approximation for a product:

Obtain the same result using DiscreteAsymptotic:

Introduced in 2020
 (12.1)