Questions tagged [asymptotics]

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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)},{\...
2
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3answers
91 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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0answers
40 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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0answers
39 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 ...
1
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2answers
50 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 ...
2
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1answer
101 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 ...
3
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1answer
85 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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1answer
66 views
21
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1answer
296 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 ...
2
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1answer
78 views

Why can we only find the asymptotic expression of the solution of the first implicit function?

Here are three implicit function equations ...
1
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2answers
77 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
108 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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0answers
19 views

AsymptoticLess function confusion

I have an algorithm which complexity is: ...
9
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1answer
274 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} \,...
2
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1answer
168 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\...
2
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0answers
129 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
94 views

Functional operations with data [closed]

I have the following data (a shorter sample of the whole data) : ...
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0answers
69 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
126 views

Finding series expansion of solution of algebraic equation

I have the following algebraic equation: ...
3
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2answers
66 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
95 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 ...