WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

Asymptotic
✖
Asymptotic

✖
Asymptotic[expr,xx0]

gives an asymptotic approximation for expr near x0.

✖
Asymptotic[expr,{x,x0,n}]

gives an asymptotic approximation for expr near x0 to order n.

Details and Options

  • Asymptotic is typically used to solve problems for which no exact solution 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.
  • Asymptotic[expr,x->x0] computes the leading term in an asymptotic expansion for expr. Use SeriesTermGoal to specify more terms.
  • The expression expr can be any function , an integral specified by Integrate, LaplaceTransform or InverseLaplaceTransform, a differential equation specified by DSolveValue, etc.
  • The expansion point x0 can be any finite or infinite real or complex number.
  • The approximation order n can be any positive integer or Infinity. If n is set to Infinity and expr is an analytic function of x, then Asymptotic returns the complete power series expansion of expr around x0. »
  • 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±∞
    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 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
    SeriesTermGoal Automaticnumber 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 Asymptotic. 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, Asymptotic 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, Asymptotic may return an asymptotic approximation with a lower precision even if the input has infinite precision.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Find the leading asymptotic approximation for Sin at 0:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-cbfen
Out[1]=1

Plot the function and the approximation:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-wad2
Out[2]=2

Use SeriesTermGoal to obtain a better approximation:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-macivq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-ja8zyr
Out[2]=2

Compute the leading term for an indefinite integral:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-bmwil6
Out[1]=1

Obtain the same result using AsymptoticIntegrate:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-bpucnr
Out[2]=2

Compute the leading term for an inverse Laplace transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-h9g69r
Out[1]=1

Compute an asymptotic expansion for a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-i8x8r3
Out[1]=1

Obtain the same result using AsymptoticDSolveValue:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-eu0cxk
Out[2]=2

Scope  (36)Survey of the scope of standard use cases

Elementary Functions  (8)

Leading asymptotic term for a polynomial near Infinity:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-ilwlnp
Out[1]=1

Plot the function and the approximation:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-nhshj
Out[2]=2

Rational functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-cb3dlh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-ct0rsb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-bmau5m
Out[3]=3

Exponential functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-d0aatg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-fdpbay
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-xgl96
Out[3]=3

Polynomial exponential functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-cgcqu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-cg0wcq
Out[2]=2

Rational exponential functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-gf5s6y
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-tfp1g
Out[2]=2

Hyperbolic functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-dw8ew2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-bte9a9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-ydttf
Out[3]=3

Logarithmic functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-ksegpo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-gdha6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-pr13qd
Out[3]=3

Q-functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-bs1dit
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-ifel25
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-7zp4x
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cf2vzojooa774a-bs8q30
Out[4]=4

Special Functions  (8)

Leading asymptotic term for Gamma:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-f2v245
Out[1]=1

Plot the function and the approximation:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-46w6v
Out[2]=2

Leading asymptotic term for a composite function involving Gamma:

In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-lz9he5
Out[3]=3

Airy functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-b9pld7
Out[1]=1

Plot the function and the approximation:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-b0pdd2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-m287vo
Out[3]=3

Bessel functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-e0cgmu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-3qjuy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-wxhls
Out[3]=3

Hypergeometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-b5pjs3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-lg9djv
Out[2]=2

Elliptic functions:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-ebhyy2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-t2yju
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-bysjnd
Out[3]=3

Functions involving HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-mu2cz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-cdps62
Out[2]=2

Functions involving PolyGamma:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-f2e5s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-dhue
Out[2]=2

Asymptotic series for QPolyGamma:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-ewr1h8
Out[1]=1

Plots of the first three approximations around :

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-binhar
Out[2]=2

Power Series Representations  (3)

Compute the power series expansion of around 0:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-qqgth0
Out[1]=1

Compute the power series expansion along with the radius of convergence:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-08hhp
Out[2]=2

Compute the power series expansion of around 0:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-c2u8lh
Out[1]=1

Obtain the first seven nonzero terms in the series:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-edj6r3
Out[2]=2

Obtain the same result directly using Asymptotic:

In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-jbetmn
Out[3]=3

Compute the power series expansion of around 1:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-fra62q
Out[1]=1

Integrals  (2)

Compute the leading term for an indefinite integral:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-5ubuf
Out[1]=1

Obtain the same result using AsymptoticIntegrate:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-fj4umf
Out[2]=2

Compute the leading term for an Inactive definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-ijd8vx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-kiafak
Out[2]=2

This gives the result for the integral from 0 to Infinity:

In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-bq9syz
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cf2vzojooa774a-exvhrj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0cf2vzojooa774a-d1z5gs
Out[5]=5

Integral Transforms  (13)

Compute the leading term for a Laplace transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-d4gzmq
Out[1]=1

Compare with a numerical approximation:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-cecua4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-dxxssa
Out[3]=3

Compute the leading term for an inverse Laplace transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-w0sef
Out[1]=1

Compute the leading term for a Mellin transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-hnihhm
Out[1]=1

Compute the leading term for an inverse Mellin transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-svdk8
Out[1]=1

Compute the leading term for a Fourier transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-lueoqm
Out[1]=1

Compute the leading term for an inverse Fourier transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-kc6zu
Out[1]=1

Compute the leading term for a Fourier sine transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-lwbzos
Out[1]=1

Compute the leading term for an inverse Fourier sine transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-emsvn
Out[1]=1

Compute the leading term for a Fourier cosine transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-epwlgs
Out[1]=1

Compute the leading term for an inverse Fourier cosine transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-ci08u8
Out[1]=1

Compute the leading term for a Hankel transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-cp4k52
Out[1]=1

Compute the leading term for an inverse Hankel transform:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-d0amxo
Out[1]=1

Compute the leading term for a convolution:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-db3vcq
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-dn3lyi
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-ogxzh
Out[3]=3

Differential Equations  (2)

Compute the leading term for the asymptotic solution of a linear differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-izpyvm
Out[1]=1

Obtain the same result using AsymptoticDSolveValue:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-jlnbi
Out[2]=2

Compute the leading term for the asymptotic solution of a DifferentialRoot:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-fr7n0w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-g0ss79
Out[2]=2

Options  (1)Common values & functionality for each option

SeriesTermGoal  (1)

By default, Asymptotic returns the leading term in the asymptotic expansion for a function:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-btgag
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-ea8iv6
Out[2]=2

Use SeriesTermGoal to obtain more terms from the expansion:

In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-cpgj6z
Out[3]=3

Applications  (6)Sample problems that can be solved with this function

Compute the leading behavior of a function near a point:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-e4vckc
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-j98tnr
Out[2]=2

Plot the function and the leading behavior:

In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-io9y3
Out[3]=3

Compare the leading behavior of a function at 0 and Infinity:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-gaxmmc
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-fppp4x
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-zoj93
Out[3]=3

Plot the function and the leading behaviors:

In[4]:=4
✖
https://wolfram.com/xid/0cf2vzojooa774a-g6b75y
Out[4]=4

Obtain an asymptotic expansion for a definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-gj1bz0
Out[1]=1

Compare with a numerical approximation:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-bc6x0n
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-bqohqc
Out[3]=3

Find the leading behavior of the solution for a differential equation for large values of :

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-gvp5s2
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-f0l72k
Out[2]=2

Find the leading behavior:

In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-ewqoc6
Out[3]=3

Plot the solution along with the leading term:

In[4]:=4
✖
https://wolfram.com/xid/0cf2vzojooa774a-c1257w
Out[4]=4

Find the leading term of the expansion for a meromorphic function at :

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-fgspef
Out[1]=1

Find the leading term at the origin:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-dmdxkv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-b9r6nk
Out[3]=3

The prime number theorem states that is an asymptotic approximation to the prime-counting function TemplateBox[{x}, PrimePi]:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-iblwu
Out[1]=1

Compare the prime-counting function and the two approximations:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-buobdr
Out[2]=2

Properties & Relations  (6)Properties of the function, and connections to other functions

Asymptotic returns a result that is asymptotically equivalent to the input:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-k2oh0j
Out[1]=1

Use AsymptoticEquivalent to verify the result:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-oh2tg2
Out[2]=2

The result from Asymptotic is equal to the limit at the point if it exists:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-f3lm9s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-hjffhr
Out[2]=2

Asymptotic typically gives the leading term in the series expansion:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-irg6a
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-qnixb
Out[2]=2

Asymptotic computes approximations for functions of a continuous variable:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-d6ouex
Out[1]=1

DiscreteAsymptotic computes approximations for functions of a discrete variable:

In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-gphe2
Out[2]=2

Use AsymptoticExpectation to find an asymptotic approximation of an expectation:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-itsdg9
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-eqk7tf
Out[2]=2

Obtain the asymptotic approximation using Asymptotic:

In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-nv3zd8
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cf2vzojooa774a-eubzot
Out[4]=4

Use AsymptoticProbability to find an asymptotic approximation of a probability:

In[1]:=1
✖
https://wolfram.com/xid/0cf2vzojooa774a-edf4wp
In[2]:=2
✖
https://wolfram.com/xid/0cf2vzojooa774a-nisnh
Out[2]=2

Obtain the asymptotic approximation using Asymptotic:

In[3]:=3
✖
https://wolfram.com/xid/0cf2vzojooa774a-95jlg
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cf2vzojooa774a-dfiedo
Out[4]=4
Wolfram Research (2020), Asymptotic, Wolfram Language function, https://reference.wolfram.com/language/ref/Asymptotic.html (updated 2022).
✖
Wolfram Research (2020), Asymptotic, Wolfram Language function, https://reference.wolfram.com/language/ref/Asymptotic.html (updated 2022).

Text

Wolfram Research (2020), Asymptotic, Wolfram Language function, https://reference.wolfram.com/language/ref/Asymptotic.html (updated 2022).

✖
Wolfram Research (2020), Asymptotic, Wolfram Language function, https://reference.wolfram.com/language/ref/Asymptotic.html (updated 2022).

CMS

Wolfram Language. 2020. "Asymptotic." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Asymptotic.html.

✖
Wolfram Language. 2020. "Asymptotic." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Asymptotic.html.

APA

Wolfram Language. (2020). Asymptotic. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Asymptotic.html

✖
Wolfram Language. (2020). Asymptotic. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Asymptotic.html

BibTeX

@misc{reference.wolfram_2025_asymptotic, author="Wolfram Research", title="{Asymptotic}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Asymptotic.html}", note=[Accessed: 25-March-2025 ]}

✖
@misc{reference.wolfram_2025_asymptotic, author="Wolfram Research", title="{Asymptotic}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Asymptotic.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_asymptotic, organization={Wolfram Research}, title={Asymptotic}, year={2022}, url={https://reference.wolfram.com/language/ref/Asymptotic.html}, note=[Accessed: 25-March-2025 ]}

✖
@online{reference.wolfram_2025_asymptotic, organization={Wolfram Research}, title={Asymptotic}, year={2022}, url={https://reference.wolfram.com/language/ref/Asymptotic.html}, note=[Accessed: 25-March-2025 ]}