0
votes
0answers
17 views

Existence of inverse Laplace tranform

I have two questions about inverse Laplace transform. Given a function $F(s)$, does its inverse Laplace tranform always exists ? If it's not, assume $F(s)$ has an inverse Laplace tranform, does the ...
2
votes
0answers
34 views

Laplace transform to describe a bounded function

It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by ...
4
votes
3answers
183 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
1
vote
0answers
77 views

Laplace transform - frequency differentiation property (generalization)

Let $\mathcal{L(f(t);s)}$ be the Laplace transform of a function $f$. It is known that the Laplace transform of $\mathcal{L}{(t^nf(t);s)}$ is given as (frequency differentiation property) ...
0
votes
1answer
69 views

If $f$ has a Laplace transform $F$ then $\lim_{s\to\infty}F(s)=0$?

As I know, well-known functions that have Laplace transform vanishes at infinity. Because almost well-known functions has exponential form and the Laplace transform of function has exponential form ...
0
votes
0answers
44 views

laplace transform and infinitely differentiation

This fact appears in my statistics textbook (Pg 543, statistical decision theory and bayesian analysis). it says : for normal distribution the generalized bayes estimator becomes \begin{align*} ...
0
votes
0answers
30 views

Laplacian for Radon inversion theorem

Can someone check my proof in regards to the inversion of the Radon transform in $\mathbb{R}^2$ and $\mathbb{R}^n$. define $(-\Delta)^a f(x) = \int_{\mathbb{R}^d} (2\pi|\xi|)^{2a} \hat{f}(\xi)e^{2\pi ...
1
vote
0answers
74 views

Inverse Laplace transform of functions with jump discontinuities

Given a function $F(s)$, suppose we define its inverse Laplace transform as: \begin{equation} f(t) = \lim_{k \to \infty} \frac{(-1)^{k}}{k!}\left(\frac{k}{t}\right)^{k+1}F^{(k)}\left( \frac{k}{t} ...
2
votes
1answer
74 views

inverse Radon transform

I'm having a trouble understanding a proof in regards to the inversion of the Radon transform in $\mathbb{R}^3$. The statement is as follows: if $f \in \mathcal{S}(\mathbb{R^3})$, then ...
0
votes
1answer
42 views

Analyticity of a two-sided Laplace-Stieltjes transform

Consider $$ g(y)=\int_{-\infty}^{+\infty} e^{-yt}d\mu(t) $$ convergent for $y\in(a,b)$ for some $a,b>0$; and with $\mu(t)$ a $\sigma$-finite and non-negative Borel measure on $\mathbb{R}$. I'm ...
0
votes
1answer
69 views

Analytic Function vs Exponential Order function

We say that a function $f$ is of exponential order $\alpha$ if there exist constants: $M$, $\alpha$, $T$ such that for $x>T$ $$f(x)<M\cdot e^{\alpha x}$$ Polynomials are of exponential order. ...
3
votes
2answers
350 views

Continuity of integral function

How to show that the following function is right continuous at $0$ (that is, when $a\to0+$): $I(a) = \int_0^{\infty}\frac{\sin x}{x}e^{-ax}dx$? I know that Lebesgue integral $I(0) = \frac{\pi}{2}$. ...
2
votes
0answers
106 views

A rational integral with exponential denominator

Prove that: $$\int_{-\infty }^{+\infty }{\frac{{{x}^{4}}\text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
2
votes
2answers
176 views

improper integral involving $e^x$

Show that : $$\int_{-\infty }^{+\infty }{\frac{{{x}^{2}}\, \text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta \right)\ln ...
0
votes
1answer
50 views

A improper integral on expontential

Evaluate: $$\int_{0}^{\infty }{\frac{\left( 1-{{\text{e}}^{-px}} \right)\left( 1-{{\text{e}}^{-qx}} \right)\left( 1-{{\text{e}}^{-rx}} \right)}{{{\text{e}}^{x}}}}\text{d}x,\ \ \ p>0,\ q>0,\ ...
3
votes
0answers
108 views

Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
0
votes
0answers
71 views

About Laplace transform

I dont understand the following working, why the integral becomes double integral? $$\begin{align} & \ \ \ \int_{0}^{1}{{{\left( \frac{1}{\ln x}+\frac{1}{1-x} ...
3
votes
1answer
50 views

Example of a function

I am looking for an example of a function $f$ such that $\lim_{t\to x_n}f(t)=\infty$ for infinitely many points $x_n$ and for which the Laplace transform $\mathscr{L}(f)$ exists. I am sure it must be ...
0
votes
1answer
290 views

Does a piecewise-continuous function need to be defined at its points of discontinuities?

Is the following function considered piecewise-continuous?? I'm reading conflcting definitions in different places: some highlight that that the function need not be defined at the (jump/removable) ...
1
vote
1answer
97 views

Laplace transform with (real) compact support

Given the standard unilateral Laplace transform defined on $L^1(\mathbb R ^+)$ $$ \mathscr Lf(s) = \int_0^\infty e^{-st}f(t)~dt,$$ are there any functions in $L^1$ such that $\mathscr Lf$ is ...