Articles of perturbation theory

How to find asymptotic expansions of all real roots of $x \tan(x)/\epsilon=1?$

Find expansions of all the real roots of $$x\tan(x)=\epsilon?$$ (You have to consider the first root separately) It is really bothering me. If I assume $x=x_0+x_1\epsilon +x_2\epsilon^2$ and do the taylor expansion of tan(x). Then I end up with $$(x_0+x_1\epsilon +x_2\epsilon^2)[(x_0+x_1\epsilon +x_2\epsilon^2)+1/3 (x_0+x_1\epsilon +x_2\epsilon^2)^3+…]=\epsilon$$. From the above equation $x_0=0$, then no matter what value $x_1$ […]

Asymptotic expansion of an integral

I came up with a simpler example which illustrates the technical difficulty I have encountered in my work. Consider an integral depending on parameter $\epsilon$: \begin{equation} \int\limits^{\infty}_{1 + \frac{2 \epsilon^{2}}{R^{2} – \epsilon^{2}}} \frac{1}{t^{2}-1} \frac{1}{\sqrt{R^{2} – \epsilon^{2}\left(t+1\right)/\left(t-1\right)\,}}\,{\rm d}t. \end{equation} I am interested in the behaviour of this integral $\left(~\mbox{evaluated at}\ t = \infty~\right)$ as $\epsilon \to […]

Method of dominant balance and perturbation

Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$ I can’t perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant Balance for problems like this, but I don’t really know how to proceed with that method. If someone could just get me started, then […]

Perturbative solution to $x^3+x-1=0$

I would like to calculate the real solution of $$ x^3+x-1=0 $$ by resumming a perturbation series. To this end, I considered $$ x^3+\epsilon x-1=0, $$ $\epsilon$ being a perturbation parameter. The real solution of the unperturbed equation (i.e. for $\epsilon=0$) is $x=1$, so I expanded $x$ in the formal power series $$ x(\epsilon)=1+\sum_{n=1}^\infty a_n […]

Simple question on perturbation theory for a function with two small parameters

Suppose I have an analytic function $f(x;\epsilon, \delta)$ that depends on two small parameters that are of the same order, $0 < \epsilon << 1$ and $0< \delta <<1$ . For example in the context of odes, $\dot{x} = f(x; \epsilon, \delta)$, and so expanding $f$ in Taylor series, we have $f(x;\epsilon, \delta) = f(x;0,0) […]

Find the leading order uniform approximation when the conditions are not $0<x<1$

$$\epsilon y”+y’\sin x+y\sin 2x = 0$$ with boundary conditions $y(0)=\pi$ and $y(\pi)=0$ as $\epsilon \rightarrow 0$. I don’t know how to find out where the boundary layer is? I thought initially it was at $x=0$, but this just leads to the outer solution of $$y=Ae^{-2\sin x}$$ if I’ve done it correctly and applying the boundary […]

Asymptotic expansion of exp of exp

I am having difficulties trying to find the asymptotic expansion of $I(\lambda)=\int^{\infty}_{1}\frac{1}{x^{2}}\exp(-\lambda\exp(-x))\mathrm{d}x$ as $\lambda\rightarrow\infty$ up to terms of order $O((\ln\lambda)^{-2})$. How does $(\ln\lambda)^{-1}$ appear as a small parameter? Please help. Thank you.

Approximating definite integral over infinitesimal interval (reformulated)

Pursuant to helpful comments by user254433, I have decided to take another swing at this problem while reformulating it with a simplified example. (Reformulated) General Problem: Generally speaking, I am trying to evaluate a definite integral of the form $\mathcal{I}(L)\equiv \int_0^{L}dy f(y,L) $ in the limit that $L\to 0$ (for $f$ real). Now, if the […]

Asymptotic expansions for the roots of $\epsilon^2x^4-\epsilon x^3-2x^2+2=0$

I’m trying to compute the asymptotic expansion for each of the four roots to the following equation, as $\epsilon \rightarrow 0$: $\epsilon^2x^4-\epsilon x^3-2x^2+2=0$ I’d like my expansions to go up through terms of size $O(\epsilon^2)$. I´ve made the change of variables $x=\delta y$, performed dominant balance and found out that the only two valid options […]

Reference: Continuity of Eigenvectors

I am looking for an appropriate reference for the following fact. For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix), there exist $\varepsilon, L > 0$, such that for all $H \in \mathbb{R}^{n \times n}_{\text{sym}}$ with $\|H\| \le \varepsilon$ the following holds: There are orthogonal matrices $P, Q$, such that \begin{equation*} Q^\top \, X \, […]