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Dispersion
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12 votes
5 answers
2k views
+50

Evaluating the integral $\int_0^{\infty} \frac{\sin(x)}{\sinh(x)}\,dx$

8 votes
2 answers
179 views

How to justify interchange of summation and integral with $f_n(x)=x^2 \cos(2nx)/n$ on $[0, \pi/2]\times \mathbb{N}$?

6 votes
2 answers
167 views

Upper and lower bounds for $\int_0^\infty e^{-u(u^\epsilon -1)}\,du$

6 votes
1 answer
171 views

Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$

5 votes
1 answer
637 views

Contour for $\int_0^\infty \arctan(z) e^{-z^2}\,dz$ or some variant

4 votes
1 answer
1k views

Why is a differential a dual basis vector (i.e. why $dx^i \frac{\partial}{\partial x^j} =\delta^i_j$)?

4 votes
2 answers
173 views

How to simplify the result I obtain for the following integral: $\int_0^\infty \frac{\cos(a x)}{x^4+b^4}\,dx$

4 votes
3 answers
107 views

Prove the convergence of $\int_1^{\infty} x^{-x}\,dx$

4 votes
4 answers
221 views

Trying to prove that if $n^2=3q$, then $n=3p$ for $n,p,q \in\mathbb{N}$.

4 votes
2 answers
190 views

Examples of ODEs that are toy models for complicated phenomena in PDEs

4 votes
1 answer
65 views

If $f$ is Schwartz, does $f=\mathcal{O}\left(e^{-a|x|^{\epsilon}}\right)$ for some $a,\epsilon>0$?

3 votes
1 answer
115 views

A holomorphic function satisfying $f(z)=f(iz)$

3 votes
2 answers
144 views

A function in $L^1(\mathbb{R}^d)$ satisfying $f(x)+f\ast g(x)=e^{-|x|^2}.$

3 votes
1 answer
37 views

Showing that $\|f-g\|_{L^1}=\int_{-\infty}^\infty \mu\left[\left(F(t) \setminus G(t)\right) \cup \left(G(t) \setminus F(t)\right)\right] \,dt.$

3 votes
1 answer
98 views

Behavior for small and large times of $\ddot{y}(t)=y^2(t), y(0)=y_0>0, \dot{y}(0)=0$.

3 votes
0 answers
90 views

Asymptotics of integral representation of distribution

3 votes
0 answers
59 views

Question on estimate in one of Jean Bourgain's 1992 papers

3 votes
1 answer
113 views

How to compute this definite improper integral using the Residue Theorem??

3 votes
2 answers
408 views

How to derive the identity $\sin x/x=\prod_{n=1}^\infty \cos(x/2^n)$ without using telescoping?

2 votes
0 answers
33 views

What is the adjoint of $e^{\alpha \psi} \partial_{x}^4 e^{-\alpha \psi}$?

2 votes
3 answers
451 views

How do I prove the existence of uniqueness of the vector $x$? Exercise 3-1.b in Linear Algebra Done right.

2 votes
1 answer
54 views

When is $e^f$ convex for non-convex $f$?

2 votes
0 answers
43 views

When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?

2 votes
0 answers
41 views

Showing holomorphicity of $f(z)=\sum_{n\ge 1} \frac{(z-n)^{-n}}{n!}$ on $\mathbb{C}\setminus \mathbb{N}$

2 votes
1 answer
62 views

Showing that $f(\lambda x)\to f(x)$ in $L^4$

2 votes
0 answers
186 views

An entire function $f(z)$ that is real if and only if $z$ is real

2 votes
1 answer
189 views

What is the inverse Fourier transform of the japanese bracket $(1+4\pi^2|\xi|^2)^{-s/2}, s>0$ in $\mathbb{R}^d$ and how does it decay?

2 votes
2 answers
97 views

Showing $f$ has a weak derivative in $L^p$

2 votes
0 answers
37 views

Absolutely continuous and Lipschtiz $f$ has derivative in $L^\infty$

2 votes
2 answers
170 views

Computing the Lebesgue measure of the intersection of two one dimensional balls