# Questions tagged [asymptotics]

Questions involving asymptotic analysis, including growth of functions, Big-$O$ notation, Big-$\Omega$ and Big-$\Theta$ notations.

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### Given a half-space Fourier transform what asymptotic conclusion can we make about $f(x)$?

If we have a half-space Fourier transform $\tilde{f_{+}}(k)$ of $f(x)$ satisfying $\tilde{f_{+}}(k) =\mathcal{O}(k^{-3})$ for $k \to \infty$ what asymptotic conclusions can we say about $f(x)$? I'm a ...
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### Approximating $\int_0^{\pi/4}\cos(x t^2)\tan^2t \ \mathrm{d} t$?

I am trying to find the leading order term for this integral using a stationary phase approach: $\int_0^{\pi/4}\cos(x t^2)\tan^2t \ \mathrm{d} t$. Using Euler's formula, the integral is the real part ...
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### Remove logarithms from the solutions in form of Frobenius Series

A singular point $x_0$ of a homogeneous linear differential equation is said to be isolated if the coefficient functions are singular at $x_0$ and are single-valued analytic functions in a punctured ...
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### asymptotic solution of recurrence relation

can you help me to solve this recurrence equation asymptotically? my recurrence relation is $f\left(n\right)=f\left(\frac{n}{3}\right)+f\left(\frac{n}{6}\right)+\left(n\right)^{\sqrt{\log\ n}}$ I ...
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### transform a general linear second-order equation into the hypergeometric equation

Show that a general linear second-order equation with three regular singular points at 0, 1, and $\infty$ and no other singular points can be transformed into the hypergeometric equation.
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### A question in number theory based on asymptotics

I study number theory by myself and couldn't prove this question( Question number 12 , Chapter 13) Apostol Introduction to analytic number theory. Let f(n) be a multiplicative function such that if p ...
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### Question 13.11 on Apostol Introduction to analytic number theory

While self studying Number theory from Tom M Apostol Introduction To Analytic Number Theory I am struck on question 11 of chapter 13 on page 303. For an arithmetic function f(n) prove that (a) ...
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### Summation involving $O$-notation

I'm reading a proof where they conclude that $$\sum_{k=1}^{\infty} \frac{O(\ln k)}{\epsilon^2\ln^2k} = \sum_{k=1}^{\infty}o_k(1) = \infty,$$ for a $\epsilon > 0$. Here the subscript $k$ means that ...
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### Asymptotic sequence $\phi_{n,m}(\lambda)$

Consider a family of functions $$\phi_{nm}(\lambda) = \lambda^{m} e^{−n \lambda},$$ where the integers $n = 1, 2, 3, \cdot \cdot \cdot, \infty \$ and $\ m =1, 2, 3 , \cdot \cdot \cdot \ ,\ n.$ (a) ...
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### $O(x^y)$ and $O(x^{y+1})$, $O(C^y)$ and $O(C^{y+1})$

Let $x$, $y$ be variables and $C$ be a constant. How do these $O$ notations compare? $O(x^y)$ vs $O(x^{y+1})$ (with $x^{y+1} = x * x^y$) and $O(C^y)$ vs $O(C^{y+1})$ (with $C^{y+1} = C * C^y)$ My ...
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### Intuition behind $o(n)+\omega(n)+\Theta(n)=\Omega(n)$

Intuition behind $o(n)+\omega(n)+\Theta(n)=\Omega(n)$ Left hand side means set of all functions more than n + set of all functions greater than n + set of all functions equal to n ---> what does ...
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### Intuition behind $f(n)+o(f(n)) = \Theta(f(n))$

Intuition behind $f(n)+o(f(n)) = \Theta(f(n))$ I could prove this using classical definations of o and theta. Also the proof is given in proof that a function plus a lower growth function is theta ...
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### Laplace's method for integration

I am trying to compute $$I(\lambda) = \int_{0}^1 \frac{x}{\sqrt{1+x^4}}e^{\lambda x}\mathrm{d}x$$ for large, real, positive $\lambda$. I'm attempting this with Laplace's method as suggested, however ...