Questions tagged [asymptotics]
The asymptotics tag has no usage guidance.
81
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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
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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 $f(y) = y^2 \cos(y^2) C( (2/\pi)^{1/2} y)^2$, ...
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1
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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?
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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
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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 ...
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1
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117
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Can Mathematica estimate this complex function?
Mathematica has given me a function in $x,r$ given by
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2
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89
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How to preserve the order of expressions in the asymptotic expansion?
I have a complicated expression involving logarithms.
...
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2
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262
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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) ...
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AsymptoticIntegrate of a difficult integral [closed]
I would like to use AsymptoticIntegrate to address this problem: to get an asymptotic expression for:
$$\int\limits_0^\infty \frac{\sin (\sin x)}{\Gamma (x+1)} \ln ...
- 39.5k
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2
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Looking for the asymptotics of an asymptotics
I am trying to polish my second answer to this question in Mathematics Stack Exchange.
The problem is to find the asymptotics of $t$, solution of the implicit equation
$$\color{blue}{\left(1-2 x^2\...
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1
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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 ...
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3
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118
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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
...
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2
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196
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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 ...
- 123
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2
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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 ...
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1
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2
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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 ...
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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 ...
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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 ...
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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 ...
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115
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Asymptotic solve question
I am trying to find the asymptotic solution of the following differential equation-
...
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134
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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 ...
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Successive solutions using previously found [closed]
is there a way to use previous calculated values of solve?
solving equations based on asymptotic expansion
$x^2+x-\varepsilon=0$
$x=x_0+\varepsilon x_1 + \varepsilon^2 x_2 + \varepsilon^3 x_3$
...
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84
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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 $r\longrightarrow \infty$...
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1
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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:
...
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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 :
$$ g(t)= c\int_{0}^{1-t} t^{m-1}\left[(u+t)^{m}-u^{m}\right]^{n-2}(u+t)^{m-1} d u $$ ...
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1
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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
...
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“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 ...
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1
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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}
]
<...
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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
...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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^{-...
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3
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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 ...
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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 ...
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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 ...
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1
answer
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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 ...
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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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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 ...
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1
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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$,...
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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
...
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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....
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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))$$
?
I need more suitable and better code ...
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Asymptotic expansion at infinity given a branch cut
Basically, I have obtained the function $\rho (r)$ below as a result of integrating
$$\rho(r)=\int_{b_0}^{r}\frac{dx}{\sqrt{1-(b_{0}/x)^{1-q}}}$$
which results to
...
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Comparison of NDSolve and Asymptotic Output Tracking results: Problem identified
My question is a continuation of the topic:
Asymptotic Output Tracking: Code Issues
Edit: Take system of ODE for example:
$\begin{cases} \frac{dx}{dt}=H \cdot \alpha \sin(\omega t)+\alpha \omega \cos(\...
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Inconsistency in Asymptotic expansion of cylindrical functions
Context
I am interested in asymptotic behaviour of Cylindrical functions which are solution to the differential equation
$$ y''(x)+(x^2-1)y(x)=0\,. $$
I ask mathematica to find such solutions:
...
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2
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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:
$y=e^{-x_1^2}$
...
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