Questions tagged [asymptotics]

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Can Mathematica solve an ode asymptotically as x goes to infinity?

Given the following ode for $x\rightarrow\infty$: $$\left(x f(x)^3 \left(\frac{(x f^\prime)^\prime}{x}\right)^\prime\right)^\prime=0,$$ in the sense of "asymptotics", the equal sign is ...
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Asymptotic solve question

I am trying to find the asymptotic solution of the following differential equation- ...
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1 answer
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How can I calculate the asymptotic value of this function correctly?

I'm trying to reproduce the results of a paper which in one part of it, I have to calculate the asymptotic value of a function but I can't reproduce that result exactly. I will be so grateful if ...
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1 answer
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Successive solutions using previously found [closed]

is there a way to use previous calculated values of solve? solving equations based on asymptotic expansion $x^2+x-\varepsilon=0$ $x=x_0+\varepsilon x_1 + \varepsilon^2 x_2 + \varepsilon^3 x_3$ ...
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Non Linear Differential Equation Asymptotics

I wish to study the asymptotic behaviour of the following equation: $\frac{d^2 a}{dr^2} = 2 a(r) \phi(r)^2 + B_1 a(r) (1-\phi^2(r) + B_2 a(r)^2)$ $\phi(r)\longrightarrow 1$ as $r\longrightarrow \infty$...
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How to reproduce Asymptotic Bounds of Recurrences in Wolframalpha

In Wolframalpha's Examples for Recurrences, there are bunch of Asymptotic Bounds of Recurrences examples, like this It can get perfect result: ...
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The average of a random varible with pdf in the form of a parametric integral

The pdf of a random variable $T$ in the interval $(0,1)$ in a certain problem I am trying to solve is given by : $$ g(t)= c\int_{0}^{1-t} t^{m-1}\left[(u+t)^{m}-u^{m}\right]^{n-2}(u+t)^{m-1} d u $$ ...
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4 votes
1 answer
167 views

Asymptotic Solve

I am trying to solve a set of equations in Mathematica. My input is Solve[y*x - 1/x - 1/x^2 == 1 && z*x - 1/x^2 + 1/x == 2, {y, z}] and output is ...
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“Largest” symbolic common factor of an integer sequence (not simply GCD)

Suppose, I have a finite fragment of a quickly increasing sequence of integers $\{a_n\}$ that is too complex, unusual, or irregular for FindSequenceFunction to find ...
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2 votes
1 answer
153 views

How is this asymptotic expansion of an integral calculated?

I am strongly impressed by this example from New in 13 as = AsymptoticIntegrate[ (t^10 + 3) Exp[I λ (t^5 + t + 1)], {t, -2, 2}, {λ, Infinity, 2} ] <...
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Are these solutions correct using `AsymptoticDSolveValue`? Less::nord: Invalid comparison with I attempted

Should one worry about correctness of these solutions due to the messages they generate? Or can one safely ignore these messages? Example 1 ...
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3 answers
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Possible bug in asymptotic expansion of CoshIntegral and SinhIntegral at infinity

Edit Thanks to all contributors. I have filed a bug report under the ID [CASE:4876478] Original OP Consider this expansion ...
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2 answers
216 views

Check the convergence of double sum

I have the following double summations: Sum 1 : $\sum _{p=0}^{k-1} \left(\frac{\sqrt{\frac{(p+1) \Gamma \left(p+\frac{11}{4}\right)}{\Gamma (p+2)}}}{(p+2) \sqrt{\Gamma \left(\frac{11}{4}\right)}}-\sum ...
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3 votes
1 answer
234 views

AsymptoticSum does not give any output

I am trying to get leading terms in terms of $p$ of the following expression $\sum_{j = p+2}^{\infty} \frac{\sqrt{\Pi_{n=2}^{j} (1+(0.75/n)) }}{\sqrt{j}(1+j)} $. I know that this sum converges and is ...
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5 votes
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MMA does not provide the correct asymptote for an integral function

Given is the function $$f(x)=\int_0^\infty \mathbb{exp}\left(-\frac{x^2}{2t^2}-t\right)\mathbb{d}t$$ Mathematica returns for the asymptotic behavior $x\to\infty$ using ...
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5 votes
3 answers
222 views

Mathematica can't simplify asymptotic expressions containing constant symbols

I want to calculate simple asymptotic expressions involving positive constant symbols ($a > 0$), such as $$\lim_{x\to\infty} \operatorname{sech}(a x) \sim 2 e^{-a x}$$ Surprisingly, the ...
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6 votes
1 answer
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Asymptotic inversion of ExpIntegralEi function

I'm looking at the small-x and large-x asymptotic expansions of the inverse of exponential integral $E_1$ (https://dlmf.nist.gov/6.2#E1) $$\begin{array}{lll} E_1 & = & \int_z^\infty \frac{e^{-...
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2 votes
3 answers
199 views

Solution of a nonlinear equation depending on the parameter

I need to solve an equation Solve[m + x*(-1 + 2*x - Log[2*Pi]) + (-1 + 2*m - 4*x)*x*Log[x] == 0, x] It is not possible on a symbolic level. It would be ideal to ...
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1 vote
1 answer
91 views

Asymptotic Output Tracking: Compensator properties

Asymptotic Output Tracking: Code Issues The question is, rather, of a theoretical nature (practical applications can be viewed in the topic at the link). Asymptotic Output Tracking is said to be based ...
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Question on AsymptoticDSolveValue

I would like to use AsymptoticDSolveValue to solve following type of equations at infinity y''[x] + (1 - 1/x^s) y[x] == 0 where ...
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5 votes
1 answer
128 views

Asymptotic[] Doesn't Actually Compute

I ran into this problem while studying the asymptotic behavior of a probability distribution function called tao2. It computes correctly at positive infinity but doesn't actually compute at negative ...
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solving a matrix ODE

I am trying to solve an ODE which looks like this: $t^2*f'(t)+K.f(t)+t*G.f(t)=0$ for $K$ and $G$ some matrices 2*2 and $f$ is a vector of functions in variable $t$. ...
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Mathematica More Detailed Plot Around Asymptotes

I'm wanting to export some animations from Mathematica involving animating certain plots with asymptotes. At certain points in time, one of the parameters approaches a value where the function no ...
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2 votes
1 answer
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AsymptoticOutputTracking for output with boundary condition

I want to try asymptotic output tracking, but with inequality. There is a differential equation: $\frac{dx}{dt}=\frac{d}{dx}(-x^4)$ With output $y=\frac{d}{dx}(-x^4)$, The output should strive for $0$,...
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3 votes
2 answers
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Asymptotic expansion around infinity for inverse cdf of normal distribution

I'm trying to get a asymptotic expansion as $x\rightarrow\infty$ for a particular expression. I have ...
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1 vote
1 answer
87 views

AsymptoticDSolveValue fails at some value

I have the following simple code for obtaining the asymptotic behavior of $r(\rho)$ at infinity. The routine works well with $q=-1$ and $q=1/3$ but fails for the rest of the values where $q<1$ (e.g....
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5 votes
0 answers
124 views

Numerical verification of the estimate:

How to verify numerically with considerable accuracy in Mathematica the following : $$\int_2^x\dfrac{1}{z\Gamma(\sin^2[π\Gamma(z)/(2z)])}dz\sim\ln(\ln(x))$$ ? I need more suitable and better code ...
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1 answer
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Asymptotic expansion at infinity given a branch cut

Basically, I have obtained the function $\rho (r)$ below as a result of integrating $$\rho(r)=\int_{b_0}^{r}\frac{dx}{\sqrt{1-(b_{0}/x)^{1-q}}}$$ which results to ...
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1 vote
1 answer
48 views

Comparison of NDSolve and Asymptotic Output Tracking results: Problem identified

My question is a continuation of the topic: Asymptotic Output Tracking: Code Issues Edit: Take system of ODE for example: $\begin{cases} \frac{dx}{dt}=H \cdot \alpha \sin(\omega t)+\alpha \omega \cos(\...
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1 answer
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Inconsistency in Asymptotic expansion of cylindrical functions

Context I am interested in asymptotic behaviour of Cylindrical functions which are solution to the differential equation $$ y''(x)+(x^2-1)y(x)=0\,. $$ I ask mathematica to find such solutions: ...
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2 votes
0 answers
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Affine state-space: Nonlinear output

I am using a system of equations to experiment: $\begin{cases} x_1'=x_2 \\ x_2'=x_1^2-x_2+u \end{cases} $ As an output, I want to use the following non-linear output: $y=e^{-x_1^2}$ ...
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1 answer
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Finding an approximate solution to this integral: $\alpha(\phi,r,p,d)=\int_0^\infty w(z,r,p,d)Q(z,r,\phi)dz$

I'm working on a physics problem and encountered a rather complex integral for which I'm trying to find an approximate solution. The integral is of the following form: $\alpha(\phi,r,p,d)=\int_0^\...
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2 votes
1 answer
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Asymptotic Output Tracking: Code Issues

My question is a continuation of the topic Which way of solving from nonlinear control to choose?, and in the future I plan to expand this question. I want to try to apply this article https://www....
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2 votes
1 answer
156 views

Asymptotic expansion of an Euler-type integral

I am looking for an asymptotic expansion for large $x$ for the following integral: $$\int_{0}^{1} \frac{t^{ia}(1-t/2)^{-ia}}{\sqrt{1-t}} e^{ixt} dt$$ $a$ and $x$ are real and positive. I tried using ...
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2 votes
0 answers
88 views

Unexpected results with Integrate/AsymptoticIntegrate with assumptions

I am quite disturbed by the result provided by Mathematica, and I am not sure if I understand perfectly its behavior. What I want to do is to check the asymptotic form of the function $I(\tau)$ with $\...
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4 votes
2 answers
157 views

Not getting the correct asymptotic behaviour when sending a small parameter to zero

I want to solve this equation for $z$: $-\frac{2\pi^2}{\beta^2}z^2+\frac{2i\pi^2u}{\beta^2}z^3+z^4=0$ $\beta$ is a positive real constant, $u$ is a real variable ranging from $0$ to $2\pi$. The code ...
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1 vote
0 answers
90 views

How reliable is AsymptoticIntegrate?

In new mathematica 12 there is a new function AsymptoticIntegrate. However, it seems that it gives me incorrect results in some cases. To be completely honest, it ...
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2 answers
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Asymptotic behavior of $n^2-n$

I recently became familiar with the Power of Wolfram. As a part of the problem I'm working o, I need to find which of the following is closer to $n^2-n$ $O(n \sqrt n)$ or $O(n \log n)$ or $O(n^2)$. ...
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5 votes
0 answers
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How to get an asymptotic of the real-valued branch of the inverse function?

Consider function $f:\mathbb R^+\to\mathbb R^+$, defined as $f(x) = x + x^2\left(1 + \log x\right)$. I need to find an asymptotic approximation of its inverse function $f^{\small(-1)}\!:\mathbb R^+\to\...
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5 votes
0 answers
94 views

How to remove the smallest term from asymptotic expansion?

It is well-known that $e^{-1/x}\sim o(x^n)$ as $x\to 0^+$ for any $n\in\mathbb{N}$, thus if I do an asymptotic expansion for a function, say $f=1/(1-x)+e^{-1/x}$ as $x\to 0^+$, I expect to receive an ...
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1 vote
0 answers
29 views

Taking "intermediate" limits, e.g. ${\cal X}\to 0$ but ${\cal X}/\epsilon \to \infty $ as $\epsilon\to 0$ - *without* taking $\epsilon \to 0$ directly

I have an expression that I would like to take limits of but not in a conventional sense. Take for example, an expression like This expression involves intermediate variables ${\cal X(\epsilon)},{\...
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2 votes
3 answers
122 views

Asymptotic values of integrals

For an integral like $$D_{n}(x) \equiv \int_{0}^{x} \frac{t^{n}}{e^{t}-1} d t$$ The asymptotic values are given as $$D_{n}(x) \simeq\left\{\begin{array}{ll} n ! \zeta(n+1)-x^{n} e^{-x}+O\left(x^{n} e^{...
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Asymptotic Relation as Boundary Condition

I want to solve a system of non-linear second order differential equations. For some of the unknown functions there are boundary conditions that i know how to write them in Mathematica. One of the ...
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2 answers
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AsymptoticSolve for the Inverse

How can I find an asymptotic expansion for the inverse of the function $f[x]=x(1+x^{1/4})$ near $0$? I tried substituting $z=x^{1/4}$ and using AsymptoticSolve to ...
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2 votes
1 answer
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AsymptoticDSolveValue multiple solutions

I'm trying to solve the following ODE asymptotically. $$y(x)^2 y'(x)^2-\left(\sqrt{2} x\right)^2 y'(x)^2+y(x)^2=0$$ From ...
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3 votes
1 answer
89 views

Does AsymptoticSum work with Arithmetical Number Theoretic Functions?

The recent function AsymptoticSum works as follows: AsymptoticSum[1/k, {k, 1, n}, n -> \[Infinity]] with expected result: ...
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1 answer
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Using the solution of AsymptoticSolve into another function

...
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22 votes
1 answer
479 views

Series vs Asymptotic in 12.1

The functionality of Series and Asymptotic (new in V12.1) is very similar. In fact, they are both listed in the Asymptotics ...
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2 votes
1 answer
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Why can we only find the asymptotic expression of the solution of the first implicit function?

Here are three implicit function equations ...
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1 vote
2 answers
120 views

Determinant of matrix with asymptotic expansion

i have determinant which each element have asymptotic expansion. $\begin{bmatrix}1+5/s+6/s^3+O[1/s^4] & 1+8/s+4/s^2+O[1/s^4]\\1+2/s+2/s^3+O[1/s^4] & 1-1/s+8/s^3+O[1/s^4]\end{bmatrix}$ ...
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