# Tag Info

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 ...
• 4,613

### 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 $n$. ...
• 22.3k

### 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 ...
• 2,635
1 vote

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 $\xi(z(t)+w(... • 115k 3 votes Accepted ### Bound on$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$p,q$... • 22.3k 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 $$e^n\, n^{-n}\,\Gamma (n) \left(1-\frac{\Gamma (... • 1,412 1 vote ### Solving 'impossible' integrals with a new (?) trick For the pleasure of working an interesting integral.$$I=\int\sqrt{\tan{\left(\frac{\csc^{-1}(x)}{2}\right)}}\,dxx=\csc (2 t)\quad \implies \quad dx=-2 \cot (2 t) \csc (2 t)\,dtI=-2 \int \... • 1,412 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) = \... • 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 \...
• 4,451
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 \$...