--- title: "AsymptoticIntegrate" language: "en" type: "Symbol" summary: "AsymptoticIntegrate[f, x, x -> x0] computes an asymptotic approximation of the indefinite integral \\[Integral]f(x) \\[DifferentialD]x for x centered at x0. AsymptoticIntegrate[f, {x, a, b}, \\[Alpha] -> \\[Alpha]0] computes an asymptotic approximation of the definite integral \\[Integral]_a(\\[Alpha])^\\ b(\\[Alpha)]f(x, \\[Alpha]) \\[DifferentialD]x for \\[Alpha] centered at \\[Alpha]0. AsymptoticIntegrate[f, ..., {\\[Xi], \\[Xi]0, n}] computes the asymptotic approximation to order n." keywords: - asymptotic expansion - series expansion - perturbation expansion - approximate integration - steepest descent - stationary phase - Laplace integral - Fourier integral - exponential integral - oscillatory integral - integral representation - Laplace method - method of stationary phase - method of steepest descent - repeated integration by parts - Watson's lemma - Gaussian integral - divergent series - Taylor series - Fresnel integral - Airy integral - gamma integral canonical_url: "https://reference.wolfram.com/language/ref/AsymptoticIntegrate.html" source: "Wolfram Language Documentation" related_guides: - title: "Asymptotics" link: "https://reference.wolfram.com/language/guide/Asymptotics.en.md" - title: "Calculus" link: "https://reference.wolfram.com/language/guide/Calculus.en.md" related_functions: - title: "Asymptotic" link: "https://reference.wolfram.com/language/ref/Asymptotic.en.md" - title: "Integrate" link: "https://reference.wolfram.com/language/ref/Integrate.en.md" - title: "AsymptoticLess" link: "https://reference.wolfram.com/language/ref/AsymptoticLess.en.md" - title: "Series" link: "https://reference.wolfram.com/language/ref/Series.en.md" - title: "NIntegrate" link: "https://reference.wolfram.com/language/ref/NIntegrate.en.md" - title: "AsymptoticDSolveValue" link: "https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.en.md" - title: "AsymptoticSum" link: "https://reference.wolfram.com/language/ref/AsymptoticSum.en.md" - title: "AsymptoticExpectation" link: "https://reference.wolfram.com/language/ref/AsymptoticExpectation.en.md" --- # AsymptoticIntegrate AsymptoticIntegrate[f, x, x -> x0] computes an asymptotic approximation of the indefinite integral $\int f(x) \, dx$ for x centered at x0. AsymptoticIntegrate[f, {x, a, b}, α -> α0] computes an asymptotic approximation of the definite integral $\int_{a(\alpha )}^{b(\alpha )} f(x,\alpha ) \, dx$ for α centered at α0. AsymptoticIntegrate[f, …, {ξ, ξ0, n}] computes the asymptotic approximation to 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. * ``AsymptoticIntegrate[f, …, x -> x0]`` computes the leading term in an asymptotic expansion for the integral 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], x -> x0]`` or ``g[x] - gn[x]∈o[gn[x] - gn - 1[x]]`` as ``x -> x0``. [image] * The asymptotic approximation ``gn[x]`` is often given as a sum ``gn[x] == ∑k = 1nαkϕk[x]``, where ``{ϕ1[x], …, ϕn[x]}`` is an asymptotic scale ``ϕ1[x]≻ϕ2[x]≻⋯ > ϕn[x]`` as ``x -> x0``. Then ``AsymptoticLess[g[x] - gn[x], ϕn[x], x -> x0]`` or ``g[x] - gn[x]∈o[ϕn[x]]`` as ``x -> x0``. * Common asymptotic scales include: | | | | --- | --- | | $\left(x-x_0\right){}^0\succ \left(x-x_0\right){}^1\succ \left(x-x_0\right){}^2\succ \cdots$ | Taylor scale when x -> x0 | | $\left(x-x_0\right){}^{-3}\succ \left(x-x_0\right){}^{-2}\succ \left(x-x_0\right){}^{-1}\succ \cdots$ | Laurent scale when x -> x0 | | $x^{-1}\succ x^{-2}\succ x^{-3}\succ \cdots$ | Laurent scale when x -> ±∞ | | $\left(x-x_0\right){}^{1/p}\succ \left(x-x_0\right){}^{2/p}\succ \left(x-x_0\right){}^{3/p}\succ \ldots$ | Puiseux scale when x -> x0 | * 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: | | | | | -------------------- | ----------------- | ------------------------------------------------------------------ | | 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 | | SeriesTermGoal | Automatic | number of terms in the approximation | | WorkingPrecision | Automatic | the precision used in internal computations | * 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 (37) ### Basic Examples (3) Compute a leading asymptotic approximation for an integral: ```wl In[1]:= AsymptoticIntegrate[Sin[t x], {t, 0, 1}, x -> 0] Out[1]= (x/2) ``` --- Compute an asymptotic expansion for an integral: ```wl In[1]:= AsymptoticIntegrate[E ^ (t x), {t, 0, 1}, {x, 0, 7}] Out[1]= 1 + (x/2) + (x^2/6) + (x^3/24) + (x^4/120) + (x^5/720) + (x^6/5040) + (x^7/40320) ``` Compare with an expansion of the exact result: ```wl In[2]:= Integrate[E ^ (x t), {t, 0, 1}] Out[2]= (-1 + E^x/x) In[3]:= Series[%, {x, 0, 7}]//Normal Out[3]= 1 + (x/2) + (x^2/6) + (x^3/24) + (x^4/120) + (x^5/720) + (x^6/5040) + (x^7/40320) ``` --- Obtain the leading term in the expansion for a Gaussian integral: ```wl In[1]:= f[x_] := E^-ω (x^2 - 2 x) In[2]:= Plot[f[x] /.  {ω -> 3}, {x, 0, 7}, PlotRange -> All, Filling -> Axis] Out[2]= [image] ``` Compute the required term: ```wl In[3]:= AsymptoticIntegrate[f[x], {x, 0, ∞}, {ω, ∞, 1}] Out[3]= E^ω Sqrt[π] Sqrt[(1/ω)] ``` Compare with the result from a numerical approximation: ```wl In[4]:= % /.  {ω -> 300.} Out[4]= 1.987736808470599*^129 In[5]:= NIntegrate[Exp[-300 (x^2 - 2 x)], {x, 0, ∞}] Out[5]= 1.9877368084704693*^129 ``` ### Scope (21) #### Indefinite Integrals (3) Compute an asymptotic expansion for a trigonometric integral about ``x = 0`` : ```wl In[1]:= AsymptoticIntegrate[Sin[x], x, {x, 0, 8}] Out[1]= -1 + (x^2/2) - (x^4/24) + (x^6/720) - (x^8/40320) ``` Compare with the result given by ``Integrate`` : ```wl In[2]:= Integrate[Sin[x], x] Out[2]= -Cos[x] In[3]:= Series[%, {x, 0, 8}]//Normal Out[3]= -1 + (x^2/2) - (x^4/24) + (x^6/720) - (x^8/40320) ``` --- Asymptotic expansion for a power tower integral about ``x = 0`` : ```wl In[1]:= AsymptoticIntegrate[x^x^x, x, {x, 0, 4}] Out[1]= (x^2/2) + (1/8) x^4 Log[x]^4 + (1/27) x^3 (2 - 6 Log[x] + 9 Log[x]^2) ``` Estimate the value at a point: ```wl In[2]:= % /. {x -> 0.0005} Out[2]= 1.276537102791476`*^-7 ``` Compare with the value given by ``NIntegrate`` : ```wl In[3]:= NIntegrate[x ^ x ^ x, {x, 0, 0.0005}] Out[3]= 1.276538609502819`*^-7 ``` --- Compute an asymptotic expansion of an indefinite integral with respect to a parameter: ```wl In[1]:= AsymptoticIntegrate[E ^ (-m Sin[x]), x, {m, 0, 3}] Out[1]= x + m Cos[x] + (1/72) m^3 (9 Cos[x] - Cos[3 x]) + (1/8) m^2 (2 x - Sin[2 x]) ``` #### Exponential Integrals (6) Compute an asymptotic expansion for a Laplace transform integral: ```wl In[1]:= AsymptoticIntegrate[Exp[ -x t] / (1 + t), {t, 0, Infinity}, {x, Infinity, 7}] Out[1]= (720/x^7) - (120/x^6) + (24/x^5) - (6/x^4) + (2/x^3) - (1/x^2) + (1/x) ``` Compare with the result given by ``LaplaceTransform`` : ```wl In[2]:= LaplaceTransform[1 / (1 + t), t, x] Out[2]= -E^x ExpIntegralEi[-x] In[3]:= Series[%, {x, Infinity, 7}]//Normal//Simplify//Expand Out[3]= (720/x^7) - (120/x^6) + (24/x^5) - (6/x^4) + (2/x^3) - (1/x^2) + (1/x) ``` --- Define a function with a Gaussian kernel having a maximum at ``t = -1`` : ```wl In[1]:= f[t_] := E^-x (t^2 + t) (1 + t)^5 / 2 In[2]:= Plot[f[t] /. {x -> 3.}, {t, -3, 2}, PlotRange -> All, Filling -> Axis] Out[2]= [image] ``` Compute the leading term in the asymptotic approximation for the integral over ``{0, 2}`` : ```wl In[3]:= AsymptoticIntegrate[f[t], {t, 0, 2}, {x, Infinity, 1}] Out[3]= (1/x) ``` Compare the result with a numerical approximation: ```wl In[4]:= % /. {x -> 3000.} Out[4]= 0.000333333 In[5]:= NIntegrate[Exp[-3000(t ^ 2 + t )] (1 + t) ^ (5 / 2), {t, 0, 2}] Out[5]= 0.000333389 ``` --- Define a function with a Gaussian kernel having a maximum at ``t = 0`` : ```wl In[1]:= f[t_] := Exp[-x t ^ 2] / (t + 2) In[2]:= Plot[f[t] /. {x -> 30.}, {t, -2, 2}, PlotRange -> All, Filling -> Axis] Out[2]= [image] ``` Compute an asymptotic approximation for the integral over ``{0, 2}`` : ```wl In[3]:= AsymptoticIntegrate[f[t], {t, 0, 2}, {x, Infinity, 7}] Out[3]= (15 Sqrt[π]/2048 x^7 / 2) - (1/64 x^3) + (3 Sqrt[π]/256 x^5 / 2) - (1/32 x^2) + (Sqrt[π]/32 x^3 / 2) - (1/8 x) + (Sqrt[π]/4 Sqrt[x]) ``` Compare the result with a numerical approximation: ```wl In[4]:= % /. {x -> 300.} Out[4]= 0.0251768 In[5]:= NIntegrate[Exp[ -300 t ^ 2] / (t + 2), {t, 0, 2}] Out[5]= 0.0251768 ``` --- Define a function with an exponential kernel having a maximum at ``t = 1`` : ```wl In[1]:= f[t_] := Exp[-x(t + (1 / t))] / (t + 1) In[2]:= Plot[f[t] /. {x -> 3.}, {t, 1, 2}, Filling -> Axis] Out[2]= [image] ``` Compute the leading term in the asymptotic approximation for the integral over ``{1, 2}`` : ```wl In[3]:= AsymptoticIntegrate[f[t], {t, 1, 2}, {x, Infinity, 1}] Out[3]= (1/4) E^-2 x Sqrt[π] Sqrt[(1/x)] ``` Compare the result with a numerical approximation: ```wl In[4]:= % /. {x -> 3.} Out[4]= 0.000634143 In[5]:= NIntegrate[Exp[-3(t + (1 / t))] / (t + 1), {t, 1, 2}] Out[5]= 0.000650336 ``` --- Define a function with an exponential kernel having a maximum at ``t = 1`` : ```wl In[1]:= f[t_] := Exp[-x (t ^ 2 - 2t)] / (t + 1) In[2]:= Plot[f[t] /. {x -> 3.}, {t, 0, 3}, Filling -> Axis] Out[2]= [image] ``` Compute the leading term in the asymptotic approximation for the integral over ``{0, 3}`` : ```wl In[3]:= AsymptoticIntegrate[f[t], {t, 0, 3}, {x, Infinity, 1}] Out[3]= (1/2) E^x Sqrt[π] Sqrt[(1/x)] ``` Compare the result with a numerical approximation: ```wl In[4]:= % /. {x -> 3000.} Out[4]= 1.23717183107980259088466395524`15.954589770191005*^1301 In[5]:= NIntegrate[Exp[-3000(t ^ 2 - 2t)] / (t ^ 2 + 1), {t, 0, 3}] Out[5]= 1.23727490895287338692668622935`15.954589770191005*^1301 ``` --- Define a function with an exponential kernel having a maximum at ``t = π / 2`` : ```wl In[1]:= f[t_] := Exp[x Sin[t]] In[2]:= Plot[f[t] /. {x -> 3.}, {t, 0, Pi}, Filling -> Axis] Out[2]= [image] ``` Compute the leading term in the asymptotic approximation for the integral over ``{0, 3}`` : ```wl In[3]:= AsymptoticIntegrate[f[t], {t, 0, Pi}, {x, Infinity, 1}] Out[3]= E^x Sqrt[2 π] Sqrt[(1/x)] ``` Compare the result with a numerical approximation: ```wl In[4]:= % /. {x -> 3000.} Out[4]= 3.49925036499802522600924577936`15.954589770191005*^1301 In[5]:= NIntegrate[Exp[3000Sin[t]], {t, 0, Pi}] Out[5]= 3.49939619444446550071279865268`15.954589770191005*^1301 ``` #### Oscillatory Integrals (4) Compute an asymptotic expansion for a Fourier-type integral on $\{0,\infty \}$ : ```wl In[1]:= AsymptoticIntegrate[Exp[ I x t] / (1 + t ^ 2), {t, 0, Infinity}, {x, Infinity, 7}] Out[1]= (720 I/x^7) + (24 I/x^5) + (2 I/x^3) + (I/x) ``` --- Find the leading-term approximation for an complex oscillatory integral on $\{0,\pi \}$ : ```wl In[1]:= f[x_] := E ^ (I ω Sin[t]) In[2]:= Plot[Re[f[x] /.  {ω -> 50}], {t, 0, π}, Filling -> Axis] Out[2]= [image] ``` Compute the required approximation: ```wl In[3]:= int = AsymptoticIntegrate[f[t], {t, 0, π}, {ω, ∞, 1}] Out[3]= (E^-(I π/4) + I ω Sqrt[2 π]/Sqrt[ω]) ``` Compare with a numerical approximation: ```wl In[4]:= NIntegrate[E ^ (I 500 Sin[t]), {t, 0, π}]//Quiet Out[4]= -0.10713 + 0.0370078 I In[5]:= int /. {ω -> 500.} Out[5]= -0.107138 + 0.032981 I ``` --- Find the leading-term approximation for a real oscillatory integral involving ``Sin`` : ```wl In[1]:= f[t_] := Sin[ x t ^ 3] / Sqrt[t + 1] In[2]:= Plot[f[t] /. {x -> 30.}, {t, 0, π / 2}, Filling -> Axis, PlotRange -> All] Out[2]= [image] ``` Compute the leading term: ```wl In[3]:= int = AsymptoticIntegrate[f[t], {t, 0, π / 2}, {x, ∞, 1}] Out[3]= (1/6) ((1/x))^1 / 3 Gamma[(1/3)] ``` Compare with a numerical approximation: ```wl In[4]:= NIntegrate[Sin[ 300 t ^ 3] / Sqrt[t + 1], {t, 0, π / 2}] Out[4]= 0.0624506 In[5]:= int /. {x -> 300.} Out[5]= 0.0666967 ``` --- Compute an asymptotic expansion for a real oscillatory integral involving ``Cos`` : ```wl In[1]:= f[t_] := Cos[ x t] / (1 + E ^ (t ^ 2)) In[2]:= Plot[f[t] /. {x -> 30.}, {t, 1, 2}, Filling -> Axis, PlotRange -> All] Out[2]= [image] ``` Compute the expansion with two terms: ```wl In[3]:= int = AsymptoticIntegrate[f[t], {t, 1, 2}, {x, ∞, 2}]//Simplify Out[3]= (2 E Cos[x] - ((1 + E) (4 E^4 (1 + E) Cos[2 x] + (1 + E^4) x (1 + E^4 - 2 (1 + E) Cos[x]) Sin[x])/(1 + E^4)^2)/(1 + E)^2 x^2) ``` Compare with a numerical approximation: ```wl In[4]:= NIntegrate[Cos[ 300 t] / (1 + E ^ t ^ 2), {t, 1, 2}] Out[4]= 0.00089959 In[5]:= int /. {x -> 300.} Out[5]= 0.000899589 ``` #### General Definite Integrals (4) Compute an asymptotic expansion for the integral of a rational function at ``x = 0`` : ```wl In[1]:= AsymptoticIntegrate[1 / (t ^ 2 + x ^ 2), {t, 1, 3}, {x, 0, 4}] Out[1]= (2/3) - (26 x^2/81) + (242 x^4/1215) ``` Compare with the result given by ``Integrate`` : ```wl In[2]:= Integrate[1 / (t ^ 2 + x ^ 2), {t, 1, 3}, Assumptions -> x > 0] Out[2]= (-ArcCot[x] + ArcTan[(3/x)]/x) In[3]:= Series[%, {x, 0, 4}]//Normal Out[3]= (2/3) - (26 x^2/81) + (242 x^4/1215) ``` --- Asymptotic expansion for a trigonometric integral at ``x = π / 2`` : ```wl In[1]:= AsymptoticIntegrate[Sin[t + x] ^ 5, {t, 0, π}, {x, π / 2, 3}] Out[1]= -(1/24) π (-24 + 5 π^2) + (1/4) (-8 + 5 π^2) x - (5 π x^2/2) + (5 x^3/3) ``` Compare with the result given by ``Integrate`` : ```wl In[2]:= Integrate[Sin[t + x] ^ 5, {t, 0, π}] Out[2]= (5 Cos[x]/4) - (5/24) Cos[3 x] + (1/40) Cos[5 x] In[3]:= Collect[Series[%, {x, π / 2, 3}]//Normal, x] Out[3]= π - (5 π^3/24) + (-2 + (5 π^2/4)) x - (5 π x^2/2) + (5 x^3/3) ``` --- Asymptotic expansion for a complete elliptic integral at ``x = 0`` : ```wl In[1]:= AsymptoticIntegrate[1 / Sqrt[1 - m * Sin[θ] ^ 2], {θ, 0, Pi / 2}, {m, 0, 5}] Out[1]= (π/2) + (m π/8) + (9 m^2 π/128) + (25 m^3 π/512) + (1225 m^4 π/32768) + (3969 m^5 π/131072) ``` Compare with the result given by ``Integrate`` : ```wl In[2]:= Integrate[1 / Sqrt[1 - m * Sin[θ] ^ 2], {θ, 0, Pi / 2}, Assumptions -> 0 < m < 1] Out[2]= (EllipticK[(m/-1 + m)]/Sqrt[1 - m]) In[3]:= Series[%, {m, 0, 4}]//Normal Out[3]= (π/2) + (m π/8) + (9 m^2 π/128) + (25 m^3 π/512) + (1225 m^4 π/32768) ``` --- Asymptotic expansion for an integral at ``x = -Infinity`` : ```wl In[1]:= AsymptoticIntegrate[E ^ (x t) / (1 + t ^ 2), {t, 0, Infinity}, {x, -Infinity, 6}] Out[1]= -(24/x^5) + (2/x^3) - (1/x) ``` Compare with a numerical approximation: ```wl In[2]:= % /. {x -> -30.} Out[2]= 0.0332602 In[3]:= NIntegrate[E ^ (-30 t) / (1 + t ^ 2), {t, 0, Infinity}] Out[3]= 0.0332602 ``` #### Transform Integrals (4) Asymptotic expansion for the Laplace transform of a function: ```wl In[1]:= f[t_] := 1 / (1 + t) In[2]:= Plot[f[t], {t, 0, 4}, Filling -> Axis] Out[2]= [image] In[3]:= AsymptoticIntegrate[f[t] E ^ (-s t), {t, 0, Infinity}, {s, Infinity, 5}] Out[3]= (24/s^5) - (6/s^4) + (2/s^3) - (1/s^2) + (1/s) ``` Compare with a series expansion of the exact result using ``LaplaceTransform`` : ```wl In[4]:= LaplaceTransform[f[t], t, s] Out[4]= -E^s ExpIntegralEi[-s] In[5]:= Series[%, {s, Infinity, 5}]//Normal Out[5]= (24/s^5) - (6/s^4) + (2/s^3) - (1/s^2) + (1/s) ``` --- Asymptotic expansion for the Mellin transform of a function: ```wl In[1]:= f[t_] := t / (t ^ 3 + 1) In[2]:= Plot[f[t], {t, 0, 4}, Exclusions -> None, Filling -> Axis] Out[2]= [image] In[3]:= AsymptoticIntegrate[f[t] t ^ (s - 1), {t, 0, Infinity}, {s, 0, 2}] Out[3]= (2 π/3 Sqrt[3]) - (2 π^2 s/27) + (5 π^3 s^2/81 Sqrt[3]) ``` Compare with a series expansion of the exact result using ``MellinTransform`` : ```wl In[4]:= MellinTransform[f[t], t, s] Out[4]= (1/3) π Sec[(π/6) - (π s/3)] In[5]:= Series[%, {s, 0, 2}]//Normal Out[5]= (2 π/3 Sqrt[3]) - (2 π^2 s/27) + (5 π^3 s^2/81 Sqrt[3]) ``` --- Asymptotic expansion for the Fourier cosine transform of a function: ```wl In[1]:= f[t_] := E ^ (-t ^ 2) In[2]:= Sqrt[2 / Pi] AsymptoticIntegrate[f[t] Cos[s t], {t, 0, Infinity}, {s, 0, 2}]//Expand Out[2]= (1/Sqrt[2]) - (s^2/4 Sqrt[2]) ``` Compare with a series expansion of the exact result using ``FourierCosTransform`` : ```wl In[3]:= FourierCosTransform[f[t], t, s] Out[3]= (E^-(s^2/4)/Sqrt[2]) In[4]:= Series[%, {s, 0, 2}]//Normal Out[4]= (1/Sqrt[2]) - (s^2/4 Sqrt[2]) ``` --- Asymptotic expansion for the Fourier sine transform of a function: ```wl In[1]:= f[t_] := E ^ (-t ^ 2 + t) In[2]:= afs = Sqrt[2 / Pi] AsymptoticIntegrate[f[t] Sin[s t], {t, 0, Infinity}, {s, 0, 2}]//FullSimplify Out[2]= -(E^1 / 4 s (-2 + GammaRegularized[-(1/2), (1/4)])/2 Sqrt[2]) ``` Compare with a series expansion of the exact result using ``FourierSinTransform`` : ```wl In[3]:= FourierSinTransform[f[t], t, s] Out[3]= -(I E^-(1/4) (I + s)^2 (-1 - Erf[(1/2) - (I s/2)] + E^I s (1 + Erf[(1/2) + (I s/2)]))/2 Sqrt[2]) In[4]:= Series[%, {s, 0, 2}, Assumptions -> s > 0]//Normal//FullSimplify Out[4]= (s (2 + E^1 / 4 Sqrt[π] (1 + Erf[(1/2)]))/2 Sqrt[2 π]) In[5]:= % - afs//FullSimplify Out[5]= 0 ``` ### Options (1) #### GeneratedParameters (1) Generate an arbitrary constant for an indefinite integral: ```wl In[1]:= AsymptoticIntegrate[Cos[x], x, {x, 0, 3}, GeneratedParameters -> C] Out[1]= x - (x^3/6) + C[1] ``` The default value for the arbitrary constant is ``0`` : ```wl In[2]:= % /. {C[1] -> 0} Out[2]= x - (x^3/6) In[3]:= AsymptoticIntegrate[Cos[x], x, {x, 0, 3}] Out[3]= x - (x^3/6) ``` ### Applications (7) Obtain an asymptotic expansion for a definite integral: ```wl In[1]:= int = AsymptoticIntegrate[1 / Sqrt[1 - ϵ t ^ 14], {t, 0, 1}, {ϵ, 0, 2}] Out[1]= 1 + (ϵ/30) + (3 ϵ^2/232) ``` Compare with a numerical approximation: ```wl In[2]:= int /. {ϵ -> 0.3} Out[2]= 1.01116 In[3]:= NIntegrate[1 / Sqrt[1 - 0.3 t ^ 14], {t, 0, 1}] Out[3]= 1.01141 ``` Improve the asymptotic approximation by increasing the number of terms: ```wl In[4]:= AsymptoticIntegrate[1 / Sqrt[1 - ϵ t ^ 14], {t, 0, 1}, {ϵ, 0, 5}] Out[4]= 1 + (ϵ/30) + (3 ϵ^2/232) + (5 ϵ^3/688) + (35 ϵ^4/7296) + (63 ϵ^5/18176) In[5]:= % /. {ϵ -> 0.3} Out[5]= 1.01141 ``` --- Find the area under the curve of $f(x)=\sin ^2(a x) \cos ^4(x)$ from $-\pi$ to $\pi$ : ```wl In[1]:= Plot[Sin[a x] ^ 2 Cos[x] ^ 4 /. {a -> 1.05}, {x, -π, π}, Filling -> Axis] Out[1]= [image] In[2]:= AsymptoticIntegrate[Sin[a x] ^ 2 Cos[x] ^ 4, {x, -π, π}, {a, 1, 4}] Out[2]= -(1/540) a^2 π (-2305 + 1500 π^2 + 108 π^4) - (a^4 π (-11765 + 8760 π^2 + 864 π^4)/25920) - (π (-48995 + 12360 π^2 + 864 π^4)/25920) + (a^3 π (-28015 + 21480 π^2 + 1728 π^4)/12960) + (a π (-56065 + 25080 π^2 + 1728 π^4)/12960) ``` Find the area for a specific value of $a$ : ```wl In[3]:= % /. {a -> 1.05} Out[3]= 0.338533 ``` --- Compute the volume enclosed when $y=\sin ^2(a x)$ for $0 3.1} Out[2]= 3.58171 ``` Visualize the solid: ```wl In[3]:= RevolutionPlot3D[Sin[3.1 x] ^ 2, {x, 0, π}, RevolutionAxis -> "X"] Out[3]= [image] ``` --- Define an asymptotic version of ``LaplaceTransform`` : ```wl In[1]:= asymLaplaceTransform[f_, t_, s_, n_] := AsymptoticIntegrate[f E ^ (-s t), {t, 0, ∞}, {s, ∞, n}] ``` Use it to compute the asymptotic Laplace transform of a periodic function: ```wl In[2]:= f[t_] := 1 / (3 + Sin[t] ^ 3) In[3]:= Plot[f[t], {t, 0, 20}, PlotRange -> All, Filling -> Axis] Out[3]= [image] In[4]:= lap = asymLaplaceTransform[f[t], t, s, 8] Out[4]= -(182/3 s^8) + (80/3 s^7) + (20/3 s^6) - (2/3 s^4) + (1/3 s) ``` Plot the result: ```wl In[5]:= Plot[lap, {s, 100, 1000}] Out[5]= [image] ``` Compare with a numerical approximation: ```wl In[6]:= Table[NIntegrate[f[t] E ^ (-s t), {t, 0, ∞}], {s, 100, 1000, 200}] Out[6]= {0.00333333, 0.00111111, 0.000666667, 0.00047619, 0.00037037} In[7]:= Table[lap, {s, 100, 1000, 200}]//N Out[7]= {0.00333333, 0.00111111, 0.000666667, 0.00047619, 0.00037037} ``` --- Compute the asymptotic mean for a ``ProbabilityDistribution`` with domain $\{0,1\}$ : ```wl In[1]:= dist = ProbabilityDistribution[1 / (1 + ϵ (x - 2 / 3) ^ 2), {x, 0, 1}, Assumptions -> ϵ > 0]; ``` Normalize the ``PDF`` : ```wl In[2]:= norm = Integrate[1 / (1 + ϵ (x - 1 / 3) ^ 2), {x, 0, 1}, Assumptions -> ϵ > 0] Out[2]= (ArcCot[(3/Sqrt[ϵ])] + ArcTan[(2 Sqrt[ϵ]/3)]/Sqrt[ϵ]) ``` Plot the PDF: ```wl In[3]:= Plot[PDF[dist, x] / norm /. {ϵ -> 1 / 20}, {x, 0, 1}, Filling -> Axis] Out[3]= [image] ``` Define the asymptotic mean: ```wl In[4]:= asympMean[dist_, ϵ_, n_] := AsymptoticIntegrate[x Refine[PDF[dist, x], 0 < x < 1], {x, 0, 1}, {ϵ, 0, n}] ``` Compute the asymptotic mean for the given distribution: ```wl In[5]:= asympMean[dist, ϵ, 3] Out[5]= (1/2) - (ϵ/36) + (ϵ^2/270) - (31 ϵ^3/40824) ``` Compare with an expansion of the exact result: ```wl In[6]:= Mean[dist] Out[6]= (4 Sqrt[ϵ] ArcCot[(3/Sqrt[ϵ])] + 4 Sqrt[ϵ] ArcTan[(2 Sqrt[ϵ]/3)] + 3 Log[(9 + ϵ/9 + 4 ϵ)]/6 ϵ) In[7]:= Series[%, {ϵ, 0, 3}, Assumptions -> ϵ > 0]//Normal Out[7]= (1/2) - (ϵ/36) + (ϵ^2/270) - (31 ϵ^3/40824) ``` --- Obtain an asymptotic expansion for the integral representation of ``BesselJ`` : ```wl In[1]:= besselint = ((2 ^ (1 - ν) * z ^ ν) / (Sqrt[Pi] * Gamma[ν + 1 / 2]))AsymptoticIntegrate[(1 - t ^ 2) ^ (ν - 1 / 2) * Cos[z * t], {t, 0, 1}, {z, 0, 2}, Assumptions -> ν > 0] Out[1]= (2^1 - ν z^ν (-(Sqrt[π] z^2 Gamma[(1/2) + ν]/8 Gamma[2 + ν]) + (Sqrt[π] (1 + ν) Gamma[(1/2) + ν]/2 Gamma[2 + ν]))/Sqrt[π] Gamma[(1/2) + ν]) ``` Compare with numerical values of the Bessel function: ```wl In[2]:= Table[besselint /. {ν -> 2}, {z, 0.1, 0.5, 0.1}] Out[2]= {0.00124896, 0.00498333, 0.0111656, 0.0197333, 0.030599} In[3]:= Table[BesselJ[2, z], {z, 0.1, 0.5, 0.1}] Out[3]= {0.00124896, 0.00498335, 0.0111659, 0.0197347, 0.030604} ``` --- Compute an asymptotic approximation for an oscillatory integral: ```wl In[1]:= f[t_] := Sin[t] E ^ (I x (t ^ 3 + 3 t)) In[2]:= ReImPlot[f[t] /. {x -> 30}, {t, -2, 2}, PlotLegends -> "ReIm"] Out[2]= [image] In[3]:= AsymptoticIntegrate[f[t], {t, -2, 2}, {x, ∞, 4}] Out[3]= ((26 I/84375 x^3) - (2 I/15 x)) Cos[14 x] Sin[2] + (-(16 I/1265625 x^4) - (8 I/1125 x^2)) Sin[2] Sin[14 x] + Cos[2] (-(8 I Cos[14 x]/5625 x^3) + (-(14 I/50625 x^4) + (2 I/225 x^2)) Sin[14 x]) ``` Compare the result with a numerical approximation: ```wl In[4]:= % /. {x -> 30.} Out[4]= 0.  - 0.0022638 I In[5]:= NIntegrate[Sin[t] E ^ (I 30 (t ^ 3 + 3 t)), {t, -2, 2}] Out[5]= 1.734723475976807`*^-18 - 0.0022638 I ``` ### Properties & Relations (4) ``AsymptoticIntegrate`` computes the integral up to a given order: ```wl In[1]:= AsymptoticIntegrate[Sin[x], x, {x, 0, 6}] Out[1]= -1 + (x^2/2) - (x^4/24) + (x^6/720) In[2]:= Series[-Cos[x] - %, {x, 0, 6}] Out[2]= SeriesData[x, 0, {}, 7, 7, 1] ``` --- Use ``Integrate`` to compute the integral in closed form: ```wl In[1]:= AsymptoticIntegrate[Sin[x t] ^ 2, {t, 0, 1}, {x, 0, 8}] Out[1]= (x^2/3) - (x^4/15) + (2 x^6/315) - (x^8/2835) In[2]:= Integrate[Sin[x t] ^ 2, {t, 0, 1}] Out[2]= (1/2) - (Sin[2 x]/4 x) In[3]:= Series[%, {x, 0, 8}]//Normal Out[3]= (x^2/3) - (x^4/15) + (2 x^6/315) - (x^8/2835) ``` --- Use ``NIntegrate`` to compute a numerical approximation: ```wl In[1]:= AsymptoticIntegrate[E ^ (-ω t ^ 2)(t + 5) ^ (1 / 2), {t, 0, 5}, {ω, ∞, 3}] Out[1]= -(Sqrt[(π/5)]/160 ω^3 / 2) + (1/4 Sqrt[5] ω) + (Sqrt[5 π]/2 Sqrt[ω]) In[2]:= % /. {ω -> 300.} Out[2]= 0.114783 In[3]:= NIntegrate[E ^ (-300 t ^ 2)(t + 5) ^ (1 / 2), {t, 0, 5}] Out[3]= 0.114783 ``` --- Use ``AsymptoticExpectation`` to compute an asymptotic expectation: ```wl In[1]:= dist = UniformDistribution[{-1, 2}]; In[2]:= AsymptoticExpectation[E ^ (a x), x\[Distributed]dist, {a, 0, 3}] Out[2]= 1 + (a/2) + (a^2/2) + (5 a^3/24) ``` Obtain the same result using ``AsymptoticIntegrate`` : ```wl In[3]:= AsymptoticIntegrate[E ^ (a x) PDF[dist, x], {x, -∞, ∞}, {a, 0, 3}] Out[3]= 1 + (a/2) + (a^2/2) + (5 a^3/24) ``` ### Possible Issues (1) The expansion returned for this example has fewer than four terms: ```wl In[1]:= AsymptoticIntegrate[Cos[x], x, {x, 0, 4}] Out[1]= x - (x^3/6) ``` The missing terms are present in the expansion for a more general integral: ```wl In[2]:= AsymptoticIntegrate[Cos[x + a], x, {x, 0, 4}] Out[2]= x Cos[a] - (1/6) x^3 Cos[a] + Sin[a] - (1/2) x^2 Sin[a] + (1/24) x^4 Sin[a] In[3]:= % /. {a -> 0} Out[3]= x - (x^3/6) ``` ## See Also * [`Asymptotic`](https://reference.wolfram.com/language/ref/Asymptotic.en.md) * [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md) * [`AsymptoticLess`](https://reference.wolfram.com/language/ref/AsymptoticLess.en.md) * [`Series`](https://reference.wolfram.com/language/ref/Series.en.md) * [`NIntegrate`](https://reference.wolfram.com/language/ref/NIntegrate.en.md) * [`AsymptoticDSolveValue`](https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.en.md) * [`AsymptoticSum`](https://reference.wolfram.com/language/ref/AsymptoticSum.en.md) * [`AsymptoticExpectation`](https://reference.wolfram.com/language/ref/AsymptoticExpectation.en.md) ## Related Guides * [`Asymptotics`](https://reference.wolfram.com/language/guide/Asymptotics.en.md) * [`Calculus`](https://reference.wolfram.com/language/guide/Calculus.en.md) ## History * [Introduced in 2018 (11.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn113.en.md) \| [Updated in 2020 (12.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn121.en.md)