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
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±∞ 
Puiseux 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:
-
AccuracyGoal Automatic digits of absolute accuracy sought Assumptions $Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate answers that involve conditions on parameters GeneratedParameters None how to name generated parameters Method Automatic method to use PerformanceGoal $PerformanceGoal aspects of performance to optimize PrecisionGoal Automatic digits of precision sought Regularization None what regularization scheme to use SeriesTermGoal Automatic number of terms in the approximation WorkingPrecision Automatic the 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
open all close allBasic Examples (2)
Compute an asymptotic approximation for a product:
Improve the approximation using SeriesTermGoal:
Compute an asymptotic expansion for a product with respect to a parameter:
Scope (14)
Basic Uses (3)
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:
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 
:
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:
Text
Wolfram Research (2020), AsymptoticProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticProduct.html.
CMS
Wolfram Language. 2020. "AsymptoticProduct." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AsymptoticProduct.html.
APA
Wolfram Language. (2020). AsymptoticProduct. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AsymptoticProduct.html