Asymptotics Definition and 19 Threads

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function f(n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n2. The function f(n) is said to be "asymptotically equivalent to n2, as n → ∞". This is often written symbolically as f(n) ~ n2, which is read as "f(n) is asymptotic to n2".
An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less than or equal to x. Then the theorem states that




π
(
x
)
∼


x

ln
⁡
x



.


{\displaystyle \pi (x)\sim {\frac {x}{\ln x}}.}

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  1. A

    A How do I apply the method of steepest descents to this integral?

    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) ##...
  2. Q

    Understand Logic of Wald & Zoupas' Expression on Conserved Quantities

    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?
  3. R

    A Asymptotic solutions to a differential equation

    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...
  4. J

    Solving Asymptotic Formula: Eq. 25 & 27

    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...
  5. mooncrater

    Different Node Deletion/Insertion in a Binary Search Tree

    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...
  6. mooncrater

    I Asymptotics for finding the successors in a Binary Tree

    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...
  7. 22990atinesh

    How to prove that (log logn)×(log log log n) = Ω(logn)

    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...
  8. V

    On nonlinearity parameter in Nonlinear Schrodinger Equation (NLS)

    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...
  9. J

    Asymptotics for a trigonometric integral

    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...
  10. S

    Who can give me some function with asymptotics ?

    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...
  11. J

    Perturbation theory and asymptotics

    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...
  12. M

    Asymptotics of a linear delay integral equation

    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...
  13. tom.stoer

    L² Hilbert space, bound states, asymptotics of wave functions

    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...
  14. tom.stoer

    L² Hilbert space, bound states, asymptotics of wave functions

    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...
  15. S

    What Are the Best Books for Asymptotic Expansions of Special Functions?

    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.
  16. D

    Using WKB(J) method to obtain asymptotics for an elastic string?

    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) +...
  17. R

    Why Does Iteration Method Give Asymptotic Solution for x = \epsilon \log(1/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...
  18. C

    Proving asymptotics to sequences

    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...
  19. C

    Primorial: asymptotics and bounds

    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 <...
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