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Looking for the asymptotic solution of $x \log (x)-(x+n) \log (x-n)=0$

Working this problem on Mathematics Stack Exchange, I very quickly obtained as an estimate of $x$ solution of $$x \log (x)-(x+n) \log (x-n)=0$$ $$x_n^{(0)}=n+\sqrt n+\frac 14 \log(n)+\frac 12 \tag 1$$ ...
Claude Leibovici's user avatar
0 votes
1 answer
67 views

Asymptotic volume of intersection of n-cube with n-sphere [closed]

I would like to symbolically compute the following: $$\lim_{d\to\infty} \frac{\text{Vol}(C^d(1) \cap B^d(K\sqrt{d}))}{\text{Vol}(C^d(1))}$$ where $B^d(r) = \{(x_1, \dots, x_d) \mid x_1^2 + \dots + x_d^...
Capybara's user avatar
2 votes
2 answers
144 views

Problems with code to solve nonlinear ODE applied to BEC

Strictly I must solve my problem with an asymptotic approximation carried out by the series method. The code I have made is the following to solve my problem $$ R''(s) + \frac{1}{s}R'(s) - \frac{1}{s^...
Litafie's user avatar
  • 33
5 votes
4 answers
410 views

Is it possible to have the asymptotics of this function?

Working the problem of $$I_n=\int_0^1 \frac{\tan ^{-1}\left(x^n\right)}{\sqrt{1-x^n}} \, dx$$ which I have not be able to compute with Mathematica. A tedious work gave the result $$I_n=\frac{\sqrt{\pi ...
Claude Leibovici's user avatar
3 votes
1 answer
84 views

How to find the asymptotic envelope of a function?

Context I am interested in the asymptotic behaviour of the envelope of a given function. Unless I missed it, it would be of interest to have a Mathematica function which when given ...
chris's user avatar
  • 23.1k
3 votes
1 answer
108 views

Asymptotic integral expansion at infinity [closed]

Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$ ...
Max's user avatar
  • 155
0 votes
0 answers
41 views

Perturbing a tensorial expression

I am new to Mathematica. I am trying to simplify an expression of the some form like: $$ n_i \sigma_{ij} n_j - \gamma n_i \hat{\sigma_{ij}} n_j = 2 + v_i x_i + \kappa E_{ij} \chi_{ji} $$ There are ...
fiarast11's user avatar
4 votes
1 answer
164 views

Series solution does not match numerical solution

I have a system of algebraic-differential equations (for background see this question): $$ 2 a'(u)=u x'(u),\\ 4 a(u)^2 \left(3 u^2 x(u)^2+1\right)=4 x(u) \big(u a(u)+4 x(u)\big)+1. $$ with initial ...
yarchik's user avatar
  • 19.2k
2 votes
1 answer
74 views

How can I investigate limit at infinity with asymptotic series?

I have the following function $h_1(x)$: ...
simon's user avatar
  • 47
2 votes
1 answer
146 views

Asymptotic solution of a system of ODEs

I have the following system of Ordinary Differential Equations (ODEs) together with initial values ...
yarchik's user avatar
  • 19.2k
1 vote
1 answer
99 views

Is the given $\lim_{x\to\infty} f(x)=0$ correct while we see the argument of cosine may become close to $\frac{\pi}2$? [closed]

I have this function $f(x)$ for integer values of $x>0$. $$f(x)=\left\lvert\frac{4}{\left(3 x^2\right) \cos \left(\sqrt{10} x\pi\right)}\right\rvert \quad \text{for}\quad x \in\mathbb N$$ I use ...
math2021's user avatar
  • 749
2 votes
1 answer
157 views

When to use Series vs Asymptotic?

I'm confused about the difference between Series and Asymptotic. Is there a good rule of thumb of when to use which? ...
Yaroslav Bulatov's user avatar
0 votes
1 answer
91 views

AsymptoticDSolveValue

this is the first time ever i am posting a problem on ANY forum as i am desperate to find asymptotic approximation to this problem with any boundary or initial condition please see if you can help ...
shahid ramji's user avatar
1 vote
0 answers
25 views

Asymptotic Solver for Nonlinear Singular Perturbations

I would like to ask if there are any built-in functions / packages that allow one to obtain asymptotic expansions for nonlinear singular perturbations. More specifically, I am dealing with ...
Dav12333's user avatar
0 votes
1 answer
121 views

Can Mathematica estimate this complex function?

Mathematica has given me a function in $x,r$ given by ...
Matthew Neil's user avatar
1 vote
2 answers
106 views

How to preserve the order of expressions in the asymptotic expansion?

I have a complicated expression involving logarithms. ...
Vaclav Kotesovec's user avatar
3 votes
2 answers
269 views

Is there any possibility of obtaining an asymptotic approximation (instead of numerical solutions) of such a 2nd-order homogeneous ODE in Mathematica?

Here is a typical linear ordinary differential equation with variable coefficients: $$\ddot{x\hspace{0pt}}(t)+\left(\frac{\ln t}t\right)^{\!2}x(t)=0\text.$$ Now I intend to investigate the (leading) ...
user688486's user avatar
0 votes
1 answer
212 views

AsymptoticIntegrate of a difficult integral [closed]

I would like to use AsymptoticIntegrate to address this problem: to get an asymptotic expression for: $$\int\limits_0^\infty \frac{\sin (\sin x)}{\Gamma (x+1)} \ln ...
David G. Stork's user avatar
5 votes
2 answers
135 views

Looking for the asymptotics of an asymptotics

I am trying to polish my second answer to this question in Mathematics Stack Exchange. The problem is to find the asymptotics of $t$, solution of the implicit equation $$\color{blue}{\left(1-2 x^2\...
Claude Leibovici's user avatar
3 votes
1 answer
105 views

Asymptotics and limits of second order ODE which depend on (two) parameters

Dear Mathematica community, for this second order ODE: h''[x] Sinh[2 x] + h'[x] 2 Cosh[2 x] - 2 h[x] Tanh[x] == 0, which is basically the harmonic equation for ...
Han's user avatar
  • 55
2 votes
0 answers
57 views

AsymptoticSolve for Taylor Expansion of Implicit Function

I have an implicit function ...
Ellen's user avatar
  • 21
3 votes
2 answers
145 views

Is there something faster than AsymptoticDSolveValue for getting the terms in a power series solution?

Here's an example of a differential equation which Mathematica 13.1 just returns without solving ...
Mr. G's user avatar
  • 335
1 vote
2 answers
259 views

AsymptoticDSolveValue returing input

I would like to find a uniform approximation of the solution to the boundary layer problem $$\epsilon y'' + (x+x^3)y'-2y=0, \hspace{20mm}y(1)=y(-1)=1$$ When using AsymptoticDSolveValue, I inputted the ...
Ron Shvartsman's user avatar
0 votes
2 answers
54 views

Prevent the use of a specific function in an output

I am trying to asymptotically integrate a function with the code AsymptoticIntegrate[Exp[I k x + I/5 k^5], {k, 0, Infinity}, x -> Infinity] and the output is ...
Ron Shvartsman's user avatar
3 votes
1 answer
75 views

Selecting the negative expression

Let Theta,t be real variables and Phi an expression of ...
Smilia's user avatar
  • 592
3 votes
0 answers
76 views

Asymptotic expansion for a function containing irrational exponents

I'm trying to find an asymptotic expansion of a function containing irrational exponents in the form of a power series (which also might contain irrational exponents). It works correctly if I request ...
Vladimir Reshetnikov's user avatar
8 votes
2 answers
416 views

Asymptotic inverse function?

A paper I am reading defines a variable $\theta$ in terms of another variable $\phi$ as an expansion in $u$, where $u$ is understood to be small: $$\theta=\phi-u^2\sin\phi+\mathcal{O}(u^4).$$ They ...
user984949's user avatar
2 votes
0 answers
139 views

Asymptotic does not give asymptotic?

Expression Asymptotic[Exp[(1 + E^x)/(1 + x)], x -> Infinity] gives Exp[(1 + E^x)/x] as the answer but these functions are ...
Anixx's user avatar
  • 3,630
1 vote
1 answer
199 views

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 ...
user95273's user avatar
  • 137
0 votes
0 answers
138 views

Asymptotic solve question

I am trying to find the asymptotic solution of the following differential equation- ...
Charlie's user avatar
  • 516
0 votes
1 answer
147 views

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 ...
Wisdom's user avatar
  • 1,278
0 votes
1 answer
67 views

Successive solutions using previously found [closed]

is there a way to use previous calculated values of solve? solving equations based on asymptotic expansion $x^2+x-\varepsilon=0$ $x=x_0+\varepsilon x_1 + \varepsilon^2 x_2 + \varepsilon^3 x_3$ ...
2Napasa's user avatar
  • 103
0 votes
0 answers
89 views

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 $r\longrightarrow \infty$...
Impala's user avatar
  • 73
2 votes
1 answer
224 views

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: ...
maplemaple's user avatar
0 votes
0 answers
32 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 : $$ g(t)= c\int_{0}^{1-t} t^{m-1}\left[(u+t)^{m}-u^{m}\right]^{n-2}(u+t)^{m-1} d u $$ ...
AgnostMystic's user avatar
4 votes
1 answer
384 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 ...
apk's user avatar
  • 307
1 vote
0 answers
77 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 ...
Vladimir Reshetnikov's user avatar
2 votes
1 answer
212 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} ] <...
user64494's user avatar
  • 27.3k
1 vote
0 answers
65 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 ...
Nasser's user avatar
  • 147k
4 votes
3 answers
223 views

Possible bug in asymptotic expansion of CoshIntegral and SinhIntegral at infinity

Bug introduced in 12.3 or earlier and persisting through 13.2 or later Edit Thanks to all contributors. I have filed a bug report under the ID [CASE:4876478] Original OP Consider this expansion ...
Dr. Wolfgang Hintze's user avatar
1 vote
2 answers
289 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 ...
honeybadger's user avatar
3 votes
1 answer
239 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 ...
honeybadger's user avatar
5 votes
0 answers
143 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 ...
granular_bastard's user avatar
5 votes
3 answers
261 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 ...
Eric Hester's user avatar
6 votes
1 answer
454 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^{-...
Yaroslav Bulatov's user avatar
2 votes
3 answers
223 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 ...
Vaclav Kotesovec's user avatar
1 vote
1 answer
97 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 ...
ayr's user avatar
  • 2,434
2 votes
0 answers
104 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 ...
user142288's user avatar
5 votes
1 answer
260 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 ...
Jade Peng's user avatar
0 votes
0 answers
87 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 $t$. ...
Joe's user avatar
  • 1