Asymptotics Definition and 19 Threads

I am trying to develop an asymptotic expansion of the following integral using the method of steepest descents: $$ \int_{0}^{\infty} \frac{1}{t+1}e^{ix(t^3-3)}dt $$ I rearranged it into the form ## \int e^{ix(t^3-3) - ln(t+1)} dt ##, where ## \phi(t) = -ln(t+1) ## and ## \psi(t)=x(t^3-3) ##...

Wald and Zoupas discussed the general definition of ``conserved quantities" in a diffeomorphism invariant theory in this work. In Section IV, they gave one expression (33) in the linked article. I cannot really understand the logic of this expression. Would you please help me with this?

I am asked to solve the differential equation $$ f''(\eta)+\frac{f'(\eta)}{\eta}+\Big(1-\frac{s^2}{\eta^2}\Big) f(\eta) - f(\eta)^3 = 0, $$ for small ##\eta## and large ##\eta## under the condition ##f(\eta \rightarrow \infty) = 1## and ##f(0)=0##. The numerically solved solution looks like...

In the following equation, $$P(x; a)= \frac{\gamma}{2\lambda L \eta} [\frac{1}{π^2N_F(a)\eta(1 - \frac{x}{a\eta})^2} + \frac{1}{π^2N_F(a)\eta(1 + \frac{x}{a\eta})^2} +\frac{2}{π^2N_F(a)\eta(1 - \frac{x^2}{a^2\eta^2})} [sin (\frac{π N_F(a)\eta(1 - \frac{x}{a\eta})^2}{2})sin (\frac{πN_F(a)\eta(1...

What happens, if instead of having any pointer pointing to a node's parent, we had pointer pointing to the node's successor? We know, that Searching would remain the same. But in my opinion Insertion and Deletion would change. This would happen, because in insertion, we would be needed to find...

Consider the algorithms : TREE-SUCCESSOR(x) 1. if x.right_child !=NIL 2. return TREE-MINIMUM(x.right_child) 3. y=x.parent 4. while y != NIL and x == y.right_child 5. x = y 6. y = y.parent 7. return y TREE-MINIMUM(x) 1. while x.left_child != NIL 2. x = x.left_child...

Is ##\log \log n \times \log \log \log n = \Omega(\log n)## How can we prove it. Actually I'm trying to prove that ##f(n) = \lceil(\log \log n)\rceil !## is polynomially bounded. It means ##c_1 n^{k_1} \leq f(n) \leq c_2 n^{k_2} \quad \forall n > n_0## ##m_1 \log n \leq \log [f(n)] \leq m_2...

While I am studying the wave propagation in fluids, the amplitude modulation seems to be governed by the Nonlinear Schrodinger (NLS) equation. In some of the journal papers the nonlinearity parameter, N seems to be of high value (N≈O(104)) and so on. I understand that weak nonlinearity...

I have become stuck while trying to evaluate the following trigonometric integral: \int_0^{\pi} (A + B \sin x)^n dx. First, I have tried to find a recurrence with respect to n, from which the closed-form solution could be calculated. However, I have failed to do this. Similarly, my effort to...

Dear Everybody, I need some some function f(r) with asymptotics as x -> 0, f-> 1*x+a*x^3+b*x^5+c*x^7+..., a,b,c>0 as x -> infinity, f-> x*Log[x] I construct one like A/B*Log[1+x], where A=1+x/2+x^3/12, B=1+d*x^2. But I think it is too complex, such that I cannot...

I need to find the roots of the transcendental function, f(x;a)=x^2-3ax-1-a+exp(-x/a)=0; I've done many problems like this before and am fairly sure this is just a regular perturbation problem. The difficulty I'm having is with the exponential term. Could anyone give me an idea of how...

Hello all, During the course of a calculation I was doing for my research, I derived a delay integral equation of the form g(x) = \int_0^1 dy K(y,x)g(x-y) where K(x,y) is a known, but somewhat ugly, kernel that has a (1-y^2)^{-1/2} singularity, but is integrable such that \int_0^1 dy...

Hi, I asked this question in the quantum physics forum https://www.physicsforums.com/showthread.php?t=406171 but (afaics) we could not figure out a proof. Let me start with a description of the problem in quantum mechanical terms and then try to translate it into a more rigorous mathematical...

Hi, I discussed this with some friends but we could not figure out a proof. Usually when considering bound states of the Schrödinger equation of a given potential V(x) one assumes that the wave function converges to zero for large x. One could argue that this is due to the requirement...

Can anyone recommend an expository book or article containing some of the derivations of the asymptotic expansions of special functions? Many of the "lookup" type books such as Abramovitz and Stegun contain the results but not the derivation of the results.

Hello, The differential equation is question that models an elastic string is: \frac {\partial^2 u}{\partial t^2} - \frac{\partial}{\partial x}\left( p(x) \frac{\partial u}{\partial x}\right)= 0 Taking u(x,t) = e^{-i\lambda t} y(x) I simplify above diff.eqn. to: \lambda^2 y(x) +...

(I don't believe this is classified as a 'homework question' since the solution is provided. Plus it's not really my homework. Apologies if I'm mistaken.) This question involves the asymptotic behaviour of x = \epsilon \log(1/x) as x tends to 0. The lecturer provided a solution using the...

Suppose I have a sequence a_0 = 1 a_n = \sum_{k=1}^n f(k)\cdot a_{n-k} where f(n) is a known function (in binomial coefficients, powers, and the like). In general, how would I go about proving that a_n\sim g(n)? I'm working on more closely estimating the function by calculating its...

I was wondering if anyone knew of good approximations of the primorial function for large numbers, or of reasonable bounds for it. By primorial I mean: n\#=\prod_{p\le n}p for p prime. All I know is n\#\sim e^n and the trivial \pi(n)!\le n\#\le n!. For small numbers (n <...