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11 votes
Accepted

Asymptotics of a strange oscillatory function

I will show (unless I made a mistake) $$ f(x) = a\sqrt{x} + O(x^{1/3}), $$ for some constant $a > 0$. For each $x > 0$, let $g(t) = \sin(x/t^2)$. Let $\varepsilon > 0$ to be chosen later ...
Matt Young's user avatar
  • 4,613
9 votes

Asymptotics of a strange oscillatory function

Yes, to (1) and (2), but the argument below is much too crude to identify $c$. The obvious attempt is to use $\sin t\ge 2t/\pi$, $0\le t\le \pi/2$ (or something similar), for the terms with large $n$. ...
Christian Remling's user avatar
8 votes

How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

I’m not sure if this is what the OP wants, but… Mathematica can integrate the indefinite integral for positive integer $n$. Inserting the upper limit $x=1$ gives zero for $n \in \mathbb N$, while the ...
Fred Hucht's user avatar
  • 2,635
1 vote

An integral involving Riemann xi function

The function $\xi$ is entire and hence continuous. Therefore and because $w(t)\to0$ and $z(t)\to a$ for some complex $a$ (as $t\to1$), we have $\xi(z(t)+1-t+w(t)e^{i \theta})\to\xi(a)$ and $\xi(z(t)+w(...
Iosif Pinelis's user avatar
3 votes
Accepted

Bound on $L^1$ norm of solution of two-point boundary value problem

First of all, there is clearly no bound if $0$ is an eigenvalue of $Lu=(pu')'+qu$, $u(0)=u(1)=0$. (This will not happen if $p(x)<0, q(x)\ge 0$ because then the operator is positive, but if $p,q$ ...
Christian Remling's user avatar
2 votes

limit of definite integral as $N \to \infty$

For an accurate calculation. Using the gamma functions $$\theta(n)=\int_0^1 (1-x)^{n-1} e^{xn} dx=e^n n^{-n} (\Gamma (n)-\Gamma (n,n))$$ Rewrite it as $$e^n\, n^{-n}\,\Gamma (n) \left(1-\frac{\Gamma (...
Claude Leibovici's user avatar
1 vote

Solving 'impossible' integrals with a new (?) trick

For the pleasure of working an interesting integral. $$I=\int\sqrt{\tan{\left(\frac{\csc^{-1}(x)}{2}\right)}}\,dx$$ $$x=\csc (2 t)\quad \implies \quad dx=-2 \cot (2 t) \csc (2 t)\,dt$$ $$I=-2 \int \...
Claude Leibovici's user avatar
10 votes

Solving 'impossible' integrals with a new (?) trick

Not a full answer, but too long for a comment. The OP's relations are related to the Gudermannian, see, e.g., https://mathworld.wolfram.com/Gudermannian.html, as $$ \operatorname{gd}(i \theta_1) = \...
Fred Hucht's user avatar
  • 2,635
1 vote
Accepted

Functions for which $\lambda f(x)=f(\alpha_\lambda x + \beta_\lambda)$

I will construct $2^\mathfrak{c}$ examples with $S=\mathbb{R}$ and $\beta_\lambda = 0$ for all $\lambda$, clearly this is the biggest we can hope for since this is how many functions $\mathbb{R}\to \...
Aleksei Kulikov's user avatar
3 votes

Behaviour of the solution of a second order ODE

A first quick answer to the question as it is (to know if $\lim_{x\to0}\dot y(x)=0$ or not), is no in general, as it is shown by the solution $y(x)=e\sqrt x$ (corresponding to $c=\frac1{\sqrt2}$ for $...
Pietro Majer's user avatar
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