**0**

votes

**0**answers

28 views

### Fourier tranform of the Euclidean norm [migrated]

where can I find the Fourier transform of the power of the Euclidean norm?, that is:
$$\mathcal{F}[\|x\|^{p}](\omega) = \int_{\mathbb{R}^{d}}\exp(-2\pi i \langle\omega, x\rangle) \|x\|^{p} dx$$
...

**6**

votes

**1**answer

213 views

### Rotation invariance of an integral

Consider the integral depending on 2 parameters
$$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$
where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...

**3**

votes

**2**answers

210 views

### Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...

**-3**

votes

**0**answers

20 views

### List comparison [on hold]

What is the best way to compare two lists? For example, how far is the list 2284, 1161, 493, 227, 116, 43, 13, 7 from being a list where each number is half the ...

**-4**

votes

**0**answers

26 views

### Calculus limit question [on hold]

enter image description here
this formula, why this must be the first item 2x ^ 3 multiplied by 2
One is -5x ^ 5 / (3x ^ 2) (4x ^ 3) another is a -5x ^ 5 / (3x ^ 2) (2x ^ 3) all can be reduction, why ...

**2**

votes

**0**answers

101 views

### Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$.
Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$
Therefore $$\sum\limits_{k=1}^\infty ...

**1**

vote

**0**answers

34 views

### convergence of ODE [closed]

I have 2 coupled linear ODEs. I used Mathematica to solve for analytical solution. But the analytical solution looks too complicated. I only need to derive some monotonicity property of the solution. ...

**2**

votes

**3**answers

216 views

### Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.
Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty ...

**2**

votes

**2**answers

88 views

### Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain:
$$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$
where $m>0$ is fixed.
Question. To ...

**0**

votes

**0**answers

126 views

### What is the value of this infinite product? [closed]

$\prod^{\infty}_{n=3}\cos\frac{\pi}{n!}=?$
I can calculate the product, $\prod^{\infty}_{n=3}\cos\frac{\pi}{2^n}$, but I don't know how to calculate the above product. Could you help me?
Thanks a ...

**1**

vote

**1**answer

99 views

### What are the best known bounds on the Hermite polynomials?

The best I could find on the net is this paper,
http://arxiv.org/pdf/math/0401310.pdf
Has this been improved?

**2**

votes

**0**answers

41 views

### Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$.
My question concerns solutions to $\triangle_g u =0$ that are say ...

**0**

votes

**0**answers

26 views

### Uniqueness and Properties of Nonlinear 2nd Order ODE with Asymptotically Constant Coefficients

I have the following differential equation:
$$ V(z) = e^z [1-\varphi(\gamma)]+(\gamma-g)V'(z)+\frac{1}{2}\kappa^2 V''(z)$$ with
$$ e^z \varphi'(\gamma) = V'(z)$$ where $\varphi(\cdot)$ is a ...

**2**

votes

**0**answers

41 views

### 1D inhomogeneous linear Schrodinger equation

I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm ...

**2**

votes

**0**answers

27 views

### A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻²
Let $f$ be ...

**5**

votes

**4**answers

377 views

### How do the roots of a polynomial change when another polynomial is added?

I need to obtain an analytical solution to an equation of the following form:
$$
(x-a)(x-b)(x-c)=d(x-e)(x-f),
$$
where $a$, $b$, $c$, $d$, $e$, and $f$ are known numbers and $x$ is the variable.
Of ...

**1**

vote

**1**answer

149 views

### find solution of complex number recurrence equation

I have the following recurrence equation:
$$(\mu\ n + \nu) f_{n} + J\Phi^{*} \sqrt{n+1}f_{n+1} + J\Phi\ \sqrt{n}f_{n-1} = 0$$
for complex numbers $f_{n}$ where $n = 0,1,2,3,...,\infty$ and complex ...

**2**

votes

**0**answers

75 views

### Is a one-dimensional unstable manifold of an ODE a union of the associated equilibrium point and two full orbits? [closed]

Consider an ordinary differential equation (ODE) system \begin{align}
\frac{dx}{dt} = f(x)
\end{align} where $x \in \mathbb{R}^n$ ($n \geq 2$) and the vector field $f$ is defined on an open subset $X$ ...

**2**

votes

**1**answer

129 views

### Proof of Euler's reflection formula via rapidly decreasing Fourier series

Story
I want to prove Euler's reflection formula by showing that
\begin{equation*}
f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s)
\end{equation*}
is constant, where $s = \sigma + it$. It's easy to see ...

**0**

votes

**0**answers

85 views

### Does this set of (structured) equations always have a solution?

Let $r_1,\ldots,r_K$ be arbitrary positive numbers.
Does
$$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in ...

**2**

votes

**1**answer

97 views

### π based on the perimeter of inscribed polygons [closed]

So, last year I got obsessed with the idea of finding a way to calculate π that wasn't already done. After reading some history, the Greek idea of measuring polygons inscribed within circles and ...

**5**

votes

**1**answer

602 views

### Is the following integral nonzero?

Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq ...

**4**

votes

**2**answers

118 views

### An English version Borok's work on finite-infinite systems of ordinary differential equations

I am looking for the English translation of the paper by V. M. Borok (originally in Russian)
The Cauchy problem for finite-infinite systems of linear differential equations. This work is about the ...

**1**

vote

**1**answer

82 views

### Estimate a Fourier Transform [closed]

I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ ...

**2**

votes

**0**answers

56 views

### What does the square root sign tells us in the wave equation? [closed]

I have been reading the paper on wave equations, and I have some confusion in notations.
Consider the initial value problem(IVP)(Wave equation):
$\frac{\partial ^2 u } {\partial t^2}(x,t) = ...

**48**

votes

**1**answer

2k views

### Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.

**35**

votes

**5**answers

1k views

### Undergraduate ODE textbook following Rota

I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...

**5**

votes

**2**answers

124 views

### Integral over the Cantor's set Hausdorff dimension

As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...

**2**

votes

**0**answers

108 views

### The boundedness of an entire function along the imaginary axis

I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...

**5**

votes

**1**answer

132 views

### Multidimensional integrals that diverge by oscillation

It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq ...

**2**

votes

**1**answer

49 views

### How to relate this summation to standard discrete cosine transformation?

The standard type III discrete cosine transformation (DCT) is defined as follows:
$${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} ...

**-1**

votes

**0**answers

34 views

### Metrics for a Wireless Sensor Network coherent model

If we describe the signal of a Wireless Sensor Network through the following coherent Gaussian model
$$ S(x,y; \boldsymbol{r}_i) = \sum_{i=1}^N \exp\left[-\frac{(x-x_i)^2+(y-y_i)^2}{2 ...

**2**

votes

**1**answer

86 views

### Lebesgue measurability of singular set

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ and $f:Q\to\mathbb{R}$ be continuous function.
Define a superdifferential of $f$ at $x\in Q$ by
$$
D^{+}f(x)=\{p\in\mathbb{R}^{d} \mid ...

**0**

votes

**2**answers

76 views

### Root of a special rational function with positive coefficients

During my research I came across the following problem:
I need to find a root of the following function:
$$\Gamma_{N}(x) = \sum\limits_{i=0}^{M}\left(\frac{\sum\limits_{n=0}^{n_F}n\ \alpha_{i,n} ...

**-1**

votes

**0**answers

25 views

### Convexity of two variables function depending in a parameter

Let $~f_V : \Omega \subset R^2 \to R$ be a function that associates to $(x\in R^2$ the value $~f_V(x)$ (which is non linear and there is no explicit formula for $f_V(x)$).
$V \in R^M$ is a parameter ...

**6**

votes

**0**answers

198 views

### Nonzero solutions to the functional ODE $f'(x)=f(f(x))$

Does $\frac{df}{dx}=f(f(x))$ have nonzero solutions? And if so, what analytic/numerical methods can be used to characterize them?

**0**

votes

**0**answers

29 views

### Interpolation functional for BV spaces?

Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...

**3**

votes

**2**answers

71 views

### Sequence of subharmonic functions on shrinking domains

Set $G_\eta:=\{(x,y)\in \mathbb{R}^2|-\eta<x<\eta, 0<y<1\}$. If $u_\eta\geq 0$ is a sequence of subharmonic functions defined on $G_\eta$ such that
$$
\int_{G_\eta}|u_\eta|^2dx\wedge ...

**1**

vote

**1**answer

70 views

### On Wazewski's theorem on system of differential inequalities

According to Springer's Encyclopedia of Math entry on differential inequalities, T. Wazewski proved in 1950 the following theorem:
Consider the system of differential inequalities given by
$$ ...

**1**

vote

**0**answers

70 views

### Finding a closed form for a certain double integral

I am working with bivariate and accurate Birnbaum-Saunders distribution to find the probability density function of a particular model for this, I would like to find a closed form for the full below:
...

**5**

votes

**0**answers

86 views

### Complex L^1 spaces; reference request

I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. ...

**6**

votes

**2**answers

148 views

### Solution set of non-unique solutions to First order ODE's

In short: What can we say about the collection of all solutions of an ODE when we don't have uniqueness?
When we teach a first course in ODE's, we look at the equation
$f:D\to \mathbb{R}, \quad ...

**1**

vote

**1**answer

61 views

### Lower Matuszewska index of positive increasing $O$-regular functions

I am not sure if this question is too specific on notations (I think the question is intuitive, but basically the only reference I know with this kind of notations is Bingham, Goldie & Teugels ...

**1**

vote

**1**answer

25 views

### Fractional sobolev regularity of a truncated function

I want to generalize the following result to fractional derivatives, specifically the fractional Laplacian.
Consider a function f which belongs to L2, and all its first order distributional ...

**1**

vote

**0**answers

60 views

### Properties of a Sobolev bound

I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...

**2**

votes

**0**answers

66 views

### Lojasiewicz's structure theorem

The Lojasiewicz structure theorem on p. 169 in the book of Krantz/Parks A primer of real analytic functions confuses me. According to the stratification property, the zeroes of a real analytic ...

**1**

vote

**1**answer

46 views

### Upper-bounding the value of a generalized Laguerre polynomial (using recurrence relation?)

I would like to produce an easily-interpretable explicit upper bound (i.e. no unspecified constants) for the function
$$
f(n) := L_n^{\left(-n-\frac{d}{2}\right)}\left(-\frac{1}{2}\right), \quad n,d ...

**1**

vote

**0**answers

23 views

### Duality of plurisubharmonic functions

Let $F$ be a cone of upper bounded upper semicontinuous functions on a compact set set $X$ containing all the constants. Let $z\in X $ and define a class of positive measure by $$M_z^F=\{ \mu : ...

**1**

vote

**0**answers

59 views

### inverse problem to resolution of the identity

Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define ...

**-1**

votes

**1**answer

48 views

### Does element-wise concavity guarantee joint concavity?

I have a function of two variables, and I have checked that along one direction (fixing another variable), it is a monotonically increasing and concave function. Whereas in another direction (fixing ...