Questions tagged [asymptotics]
The asymptotics tag has no usage guidance.
21
questions
1
vote
0answers
24 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)},{\...
2
votes
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
$$D_{n}(x) \simeq\left\{\begin{array}{ll}
n ! \zeta(n+1)-x^{n} e^{-x}+O\left(x^{n} e^{...
0
votes
0answers
40 views
AsymptoticSolve and Series not working in limit $\to \infty$. How to solve this functional polynomial relationship?
I want to solve for $J_d$ as a function of $n_d$ for $\eta\gg1$ and $\eta\gg\eta_0$ in the following equations by eliminating $\eta$ from the two equations.
\begin{equation}
J_d = J_0\cdot\left[\...
0
votes
0answers
39 views
Asymptotic Relation as Boundary Condition
I want to solve a system of non-linear second order differential equations. For some of the unknown functions there are boundary conditions that i know how to write them in Mathematica. One of the ...
1
vote
2answers
50 views
AsymptoticSolve for the Inverse
How can I find an asymptotic expansion for the inverse of the function $f[x]=x(1+x^{1/4})$ near $0$?
I tried substituting $z=x^{1/4}$ and using AsymptoticSolve to ...
2
votes
1answer
101 views
AsymptoticDSolveValue multiple solutions
I'm trying to solve the following ODE asymptotically.
$$y(x)^2 y'(x)^2-\left(\sqrt{2} x\right)^2 y'(x)^2+y(x)^2=0$$
From
...
3
votes
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:
...
0
votes
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
votes
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
vote
2answers
77 views
Determinant of matrix with asymptotic expansion
i have determinant which each element have asymptotic expansion.
$\begin{bmatrix}1+5/s+6/s^3+O[1/s^4] & 1+8/s+4/s^2+O[1/s^4]\\1+2/s+2/s^3+O[1/s^4] & 1-1/s+8/s^3+O[1/s^4]\end{bmatrix}$
...
0
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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
9
votes
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
$$\sigma_n=(1)^n\frac{\pi}{2} \big( j_{0,n+1} \,...
2
votes
1answer
168 views
Solving for the recursion relation for the expansion coefficients of the asymptotic expansion of an ODE
I want to solve for the asymptotic solution of the following differential equation
$$ \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$$
as $y\...
2
votes
0answers
129 views
Asymptotes of parabolic cylinder differential equations with boundaries at infinity
For context, I'm studying the paper Coulomb blockade in superconducting quantum point contacts by Averin from 1998. Specifically, I am trying to find how he obtains equation 11 from equation 10, which ...
-1
votes
1answer
94 views
Functional operations with data [closed]
I have the following data (a shorter sample of the whole data) :
...
0
votes
0answers
69 views
Discordance between the results of AsymptoticIntegrate and calculations
Mathematica 11.3 finds
r = Integrate[Cos[k*(x^4 - x)], {x, -Infinity, Infinity}, Assumptions -> k > 0]
...
0
votes
2answers
126 views
Finding series expansion of solution of algebraic equation
I have the following algebraic equation:
...
3
votes
2answers
66 views
Is there a way to check whether $f(x)=o(g(x))$ for given $f$ and $g$?
I would like a way to check, for two arbitrary but specified real analytic functions $f(x)$ and $g(x)$, whether $f(x)=o(g(x))$.
I am using "little-o notation," where $f(x)=o(g(x))$ is true if and ...
3
votes
1answer
95 views
AsymptoticIntegrate with multiple integration variables?
I wanted to find the asymptotic form of
$$\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
...
