Questions tagged [asymptotics]
The asymptotics tag has no usage guidance.
56
questions
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0answers
31 views
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 :
$$ g(t)= c\int_{0}^{1-t} t^{m-1}\left[(u+t)^{m}-u^{m}\right]^{n-2}(u+t)^{m-1} d u $$ ...
4
votes
1answer
127 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
...
1
vote
0answers
59 views
“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 ...
2
votes
1answer
148 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}
]
<...
1
vote
0answers
45 views
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
...
4
votes
3answers
168 views
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
...
1
vote
2answers
199 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 ...
3
votes
1answer
232 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 ...
5
votes
0answers
120 views
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 ...
5
votes
3answers
216 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 ...
6
votes
1answer
294 views
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^{-...
2
votes
3answers
192 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 ...
1
vote
1answer
90 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 ...
1
vote
0answers
57 views
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 ...
5
votes
1answer
116 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 ...
0
votes
0answers
45 views
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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0answers
50 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 ...
2
votes
1answer
40 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$,...
3
votes
2answers
158 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
...
1
vote
1answer
77 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....
5
votes
0answers
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))$$
?
I need more suitable and better code ...
1
vote
1answer
95 views
Asymptotic expansion at infinity given a branch cut
Basically, I have obtained the function $\rho (r)$ below as a result of integrating
$$\rho(r)=\int_{b_0}^{r}\frac{dx}{\sqrt{1-(b_{0}/x)^{1-q}}}$$
which results to
...
1
vote
1answer
48 views
Comparison of NDSolve and Asymptotic Output Tracking results: Problem identified
My question is a continuation of the topic:
Asymptotic Output Tracking: Code Issues
Edit: Take system of ODE for example:
$\begin{cases} \frac{dx}{dt}=H \cdot \alpha \sin(\omega t)+\alpha \omega \cos(\...
5
votes
1answer
89 views
Inconsistency in Asymptotic expansion of cylindrical functions
Context
I am interested in asymptotic behaviour of Cylindrical functions which are solution to the differential equation
$$ y''(x)+(x^2-1)y(x)=0\,. $$
I ask mathematica to find such solutions:
...
2
votes
0answers
43 views
Affine state-space: Nonlinear output
I am using a system of equations to experiment:
$\begin{cases} x_1'=x_2 \\ x_2'=x_1^2-x_2+u \end{cases} $
As an output, I want to use the following non-linear output:
$y=e^{-x_1^2}$
...
0
votes
1answer
75 views
Finding an approximate solution to this integral: $\alpha(\phi,r,p,d)=\int_0^\infty w(z,r,p,d)Q(z,r,\phi)dz$
I'm working on a physics problem and encountered a rather complex integral for which I'm trying to find an approximate solution. The integral is of the following form: $\alpha(\phi,r,p,d)=\int_0^\...
2
votes
1answer
169 views
Asymptotic Output Tracking: Code Issues
My question is a continuation of the topic Which way of solving from nonlinear control to choose?, and in the future I plan to expand this question.
I want to try to apply this article https://www....
2
votes
1answer
145 views
Asymptotic expansion of an Euler-type integral
I am looking for an asymptotic expansion for large $x$ for the following integral:
$$\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 ...
2
votes
0answers
76 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 $\...
4
votes
2answers
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 ...
1
vote
0answers
82 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 ...
0
votes
2answers
140 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)$.
...
5
votes
0answers
71 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\...
5
votes
0answers
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 ...
1
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0answers
28 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
116 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
67 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
63 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
168 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
88 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
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1answer
88 views
21
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1answer
424 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
107 views
Why can we only find the asymptotic expression of the solution of the first implicit function?
Here are three implicit function equations
...
asked Jan 15 '20 at 9:04
A little mouse on the pampas
5,2302 gold badges10 silver badges28 bronze badges
1
vote
2answers
109 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
votes
1answer
139 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 ...
0
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0answers
20 views
9
votes
1answer
310 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
$$\sigma_n=(1)^n\frac{\pi}{2} \big( j_{0,n+1} \,...
2
votes
1answer
241 views
Solving for the recursion relation for the expansion coefficients of the asymptotic expansion of an ODE
I want to solve for the asymptotic solution of the following differential equation
$$ \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
179 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
103 views
Functional operations with data [closed]
I have the following data (a shorter sample of the whole data) :
...
