Questions tagged [asymptotics]

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36 views

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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12 views

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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2answers
127 views

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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32 views

Non unique asymptotic solution of a second-order ODE

I have the following code for the series solution (via Frobenius method) of the differential equation ode around $y=\infty$. The solution and its derivative are <...
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1answer
63 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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121 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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1answer
74 views

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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1answer
39 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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1answer
78 views

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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0answers
35 views

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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1answer
63 views

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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1answer
142 views

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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1answer
130 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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0answers
58 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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2answers
154 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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80 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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2answers
108 views

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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67 views

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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88 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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27 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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3answers
102 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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43 views

AsymptoticSolve and Series not working in limit $\to \infty$. How to solve this functional polynomial relationship?

I want to solve for $J_d$ as a function of $n_d$ for $\eta\gg1$ and $\eta\gg\eta_0$ in the following equations by eliminating $\eta$ from the two equations. \begin{equation} J_d = J_0\cdot\left[\...
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49 views

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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2answers
55 views

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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1answer
122 views

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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1answer
86 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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77 views
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1answer
350 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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1answer
86 views

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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2answers
92 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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1answer
123 views

Asymptotic solution of a second-order ODE containing InverseFunction

Essentially, I have a second-order differential equation given by ode below. In order to solve it, I need to obtain an asymptotic solution where $g(x)$ must vanish at infinity which will be used after ...
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20 views

AsymptoticLess function confusion

I have an algorithm which complexity is: ...
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1answer
285 views

How could this asymptotic expansion be obtained?

I must precise that I am a very limited user of Mathematica (I can only run it from time when going at university). Working this problem, I found that $$\sigma_n=(1)^n\frac{\pi}{2} \big( j_{0,n+1} \,...
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1answer
201 views

Solving for the recursion relation for the expansion coefficients of the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation $$ \left(y^2+1\right) R''(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$ as $y\...
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0answers
155 views

Asymptotes of parabolic cylinder differential equations with boundaries at infinity

For context, I'm studying the paper Coulomb blockade in superconducting quantum point contacts by Averin from 1998. Specifically, I am trying to find how he obtains equation 11 from equation 10, which ...
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1answer
97 views

Functional operations with data [closed]

I have the following data (a shorter sample of the whole data) : ...
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70 views

Discordance between the results of AsymptoticIntegrate and calculations

Mathematica 11.3 finds r = Integrate[Cos[k*(x^4 - x)], {x, -Infinity, Infinity}, Assumptions -> k > 0] ...
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2answers
145 views

Finding series expansion of solution of algebraic equation

I have the following algebraic equation: ...
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2answers
67 views

Is there a way to check whether $f(x)=o(g(x))$ for given $f$ and $g$?

I would like a way to check, for two arbitrary but specified real analytic functions $f(x)$ and $g(x)$, whether $f(x)=o(g(x))$. I am using "little-o notation," where $f(x)=o(g(x))$ is true if and ...
3
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1answer
115 views

AsymptoticIntegrate with multiple integration variables?

I wanted to find the asymptotic form of $$\int_0^1\mathrm{d}x\int_0^1\mathrm{d}y\,\mathrm{e}^{M(x-1/2)^2+M(y-1/2)^2}$$ for $M\rightarrow\infty$. I tried ...
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3answers
192 views

Finding Asymptotics for a Series

How can I find a simple expression that's asymptotic to $\sum_{i=1}^{n-1}2^i/i$? That is, Sum[2^i/i,{i,1,n-1}]. According to https://reference.wolfram.com/...