# Tagged Questions

**1**

vote

**1**answer

139 views

### All solutions to a set of integral equations

I would like a better understanding of the set of pairs $(f_1,f_2)$ of functions $[0,1] \times [0,1] \to [0,1]$ which satisfy the following conditions:
For all $y \in [0,1]$, $f_1(x,y) \geq ...

**2**

votes

**2**answers

327 views

### How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration?
$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question here , ...

**1**

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**0**answers

58 views

### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II

This is a modification of a previous question.
Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose,
...

**6**

votes

**0**answers

264 views

### Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} ...

**4**

votes

**2**answers

224 views

### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform

Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose
$$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$
Is $a(x,0)=0$, $\forall x\in[0,1]$?
...

**1**

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**2**answers

266 views

### how to solve a singular integral equation involving the kernel $1/x$

Dear all,
Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...

**0**

votes

**0**answers

811 views

### An inverse Laplace transform involving Error function

Dear MOs,
I need to calculate the inverse Laplace transform of the following function
$$
g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0.
$$
I have checked, among many ...

**6**

votes

**0**answers

192 views

### Regularity class of certain diffeomorphisms of the real line.

I care about the following class of homeomorphisms of $\mathbb R$, which I'll call $\mathcal C^?$.
For simplicity, let us restrict attention to compactly supported homeomorphisms
(a homeomorphism ...

**8**

votes

**1**answer

394 views

### Properties of a matrix-valued generalization of the $\Gamma$ function

I am interested in the following function (Mellin transform of matrix exponential):
$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$
Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ ...

**1**

vote

**3**answers

732 views

### A definite integral

Hello,
I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might ...

**2**

votes

**2**answers

462 views

### Fonction spéciale

Est-ce qu'il y a une fonction spéciale sous la forme suivante:
$$\int e^{-st}e^{-(\alpha/(1+t))}t^{\beta-1}(1+t)^{-(\beta-1)-\gamma}dt$$
ou
$$\int ...

**15**

votes

**2**answers

907 views

### Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform.

The Hilbert transform on the real Hilbert space $L^2(\mathbb R)$ is the singular integral operator
$$
\mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy.
$$
It satisfies ...

**2**

votes

**1**answer

370 views

### Integral kernel of form $e^{-<x,y>^2}$

Let $K(x,y): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by $K(x,y) = e^{-< x,y>^2}$ where $<\cdot,\cdot>$ denote the canonical inner product. Define integral operator ...

**5**

votes

**3**answers

1k views

### When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$f(x) ...

**10**

votes

**3**answers

575 views

### MacWilliams Identity for Asymptotic Weight Spectrum of a Code

Introduction
Let $C$ be a code of block length $n$ having $A_i^C$ words of Hamming weight $i$, for $i\in [n]$, where $[n]:=\{0,\ldots,n\}$. Then, the sequence $\{ A_i^C \}_{i \in [n]} $ is called the ...

**2**

votes

**0**answers

276 views

### inverse Laplace transform of $\delta_1(\cdot)$

Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify ...