All Questions
Tagged with perturbation-theory taylor-expansion
34
questions
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29
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Expanding a function containing a logistic term inside an integral
I am working my way through Mark Homes Introduction to Perturbation Methods by myself. I am having trouble solving problem 1.8(c).
I believe that I'm supposed to split the integral by introducing a ...
- 61
1
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0
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47
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Question on perturbative expansion
I am trying to expand the experssion below in powers of the perturbative parameter $\epsilon$:
$$ \left[ \left( \frac{\omega}{k} - V(i \epsilon \partial_k) \right)^2 - f^2(k) \right] \Psi(k) $$
with ...
- 53
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286
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Taylor expansion for a Bessel function with complex argument
If we have a Bessel function of the first kind and the $m$-th order as $J_{m}(x+i\epsilon x)$, where $m$ is integer, $x, \epsilon$ are real and $\epsilon$ is a small parameter ($0<\epsilon\ll 1$), ...
- 1,309
1
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1
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144
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Prove perturbation theory breaks down
I want to find the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ terms in the pedestrian expansion $y = y_0 + \epsilon y_1 + \epsilon ^2 y_2 + \dots$, where $y$ satisfies the following second order ODE:...
- 45
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39
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How can I do the Taylor equivalent of $\sqrt{1+x^2}$ with perturbation/asymptotic expansion using $\epsilon$?
How can I do the Taylor equivalent of $\sqrt{1+x^2}$ with perturbation/asymptotic expansion using $\epsilon$?
I found some answers, but not very useful.
Expected solution would be similar to the ...
- 7,296
7
votes
1
answer
250
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How to Taylor expand a scalar quantity about a unit sphere?
Consider a vector field $\mathbf{v} (r,\theta)$ expressed in the system of (axisymmetric) spherical coordinates with $r$ denoting the radial distance and $\theta$ the polar angle.
We consider a ...
1
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0
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83
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How to determine order of the remainder term of a series for specific problems?
Background:
In the text book I have, in the 'expansion of integrands' part (perturbation theory) the author kept determining the order, but I can't understand how.
For integration of $\sin \epsilon x^...
- 131
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42
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Perturb $\tanh(c(x-x_0))$ or similar functions around $x\rightarrow x+\delta x$
Given the perturbation $x\rightarrow x+\delta x$, how can the function
$$
\tanh[c(x-x_0)],
$$
$c$ and $x_0$ constant, be perturbed? I'd like to have a form like
$$
f(x)\rightarrow f(x)+g(\delta x)
$$
...
- 113
1
vote
1
answer
195
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Why does regular perturbation fail for this example?
Consider the first order ODE:
$$ \frac{dy}{dx} = \frac{-\varepsilon y(1-x)^3-y^2(1+x)-2x(1-x)^4+x(1-x)^3}{yx(1-x)} \tag{1}$$
where $ x \in [0,\,1)$ and $\varepsilon <<1$. We substitute in the ...
- 635
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232
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'Taylor Expansion' of Integral - Asymptotic expansion - Exponential function
I need to evaluate the following integral in the limit $\kappa \ll 1$
$$\int_0^\infty exp(-\kappa t) f(t)\, dt,$$
where
$$f(x) = (1+x)(1-2x)\frac{u(x) \ln(u(x))}{u(x)^2 - 1},$$
$$u(x) = \frac{\sqrt{1+...
- 121
0
votes
0
answers
503
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Effect of perturbation on softmax
Let's say we have the following function
$$S_i(\textbf{x},T)=\frac{\exp(f_i(\textbf{x})/T)}{\sum_{j=1}^N\exp(f_j(\textbf{x})/T)}$$
where $f_{i=1,\dots,N}(\textbf{x})$ represents the output of a neural ...
- 105
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votes
1
answer
28
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Show that a wave signal superposed by a perturbed wave signal can be expressed as combined signal.
Consider the signal where $\delta > 0$ and small,
$$y(x,t)=A\sin{(kx - \omega (k) t)} + A\sin{((k + \delta)x - \omega (k + \delta) t)},$$
show that this can be rewritten as
$$y(x,t)=2A \cos{\...
- 497
2
votes
2
answers
69
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Undetermined coefficients in a perturbative expansion
In order to familiarize myself with perturbation methods, I've been trying to derive the Lorentz transformations, given by
\begin{align*}
x \rightarrow \frac{x + vt}{\sqrt{1 - v^2}} & = (x + vt)(...
- 223
2
votes
1
answer
442
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Perturbation expansion within trig function
I'm trying to find an approximate solution to a nonlinear differential equation. It involves something to the effect of
$\frac{d\Psi}{ds} = \sin{\Psi} + \dots$ , where $\Psi$ is a small variable.
If ...
- 23
1
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401
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first order Taylor expansion term of a function multiplied by a dot product of gradients.
I need to do the following Taylor expansion.
Let's suppose we have a function $f(n,|\nabla n|^2)(\nabla (|\nabla n|^2),\nabla n)$, where by $(\nabla (|\nabla n|^2),\nabla n)$ I mean the dot product ...
- 33