Questions tagged [asymptotics]

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For function f(x) = c / (3x+1) , what value of c is the 2nd function also asymptotic to (1/x) as x → ∞

for what value of c is the 2nd function also asymptotic to x1 as x → ∞ in the picture attached
Spandan Ghosh's user avatar
0 votes
0 answers
32 views

Computing Asymptotics of a Fresnel Integral At Infinity

I am very new to Mathematica, and am trying to write code that will give me the series asymptotics for the function $$ F(y) = \int_0^y f(y)\;dy, $$ where $f(y) = y^2 \cos(y^2) C( (2/\pi)^{1/2} y)^2$, ...
Jacob Denson's user avatar
2 votes
1 answer
125 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
82 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
18 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
117 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
89 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
262 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
185 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
4 votes
2 answers
131 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
96 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
  • 33
2 votes
0 answers
38 views

AsymptoticSolve for Taylor Expansion of Implicit Function

I have an implicit function ...
Ellen's user avatar
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3 votes
2 answers
118 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
196 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
45 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
64 views

Selecting the negative expression

Let Theta,t be real variables and Phi an expression of ...
Smilia's user avatar
  • 592
2 votes
0 answers
64 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
379 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
123 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
  • 3,473
0 votes
1 answer
178 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
  • 125
0 votes
0 answers
115 views

Asymptotic solve question

I am trying to find the asymptotic solution of the following differential equation- ...
Charlie's user avatar
  • 506
0 votes
1 answer
134 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
  • 1,248
0 votes
1 answer
66 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
84 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
169 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
313 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
74 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
194 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
  • 23.2k
1 vote
0 answers
60 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
  • 135k
4 votes
3 answers
215 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
261 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
235 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
133 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
249 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
407 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
218 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
94 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 ...
dtn's user avatar
  • 2,344
1 vote
0 answers
90 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
217 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
76 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
0 votes
0 answers
96 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 ...
joejoejoejoe4's user avatar
2 votes
1 answer
44 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 $0$,...
dtn's user avatar
  • 2,344
3 votes
2 answers
234 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 ...
user1936752's user avatar
1 vote
1 answer
154 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....
user583893's user avatar
5 votes
0 answers
131 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 ...
bambi's user avatar
  • 183
1 vote
1 answer
150 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 ...
user583893's user avatar
1 vote
1 answer
60 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(\...
dtn's user avatar
  • 2,344
5 votes
1 answer
104 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: ...
chris's user avatar
  • 22.5k
2 votes
0 answers
86 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: $y=e^{-x_1^2}$ ...
dtn's user avatar
  • 2,344