Linked Questions

2
votes
1answer
3k views

Fourier Transform of a Derivative [duplicate]

I'm trying to prove that: $$F\,\{f'(x)\} = -i\omega F(\omega) \qquad (1) $$ where $\, F(\omega) = F\,\{f(x)\}$ This is my procedure so far: $$F\,\{f'(x)\} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{...
24
votes
5answers
4k views

Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$

I'm trying to prove the following claim: If $f$ is continuous and $\displaystyle\int_a^\infty |f(x)|\;dx$ is finite then $\lim\limits_{ x \to \infty } f(x)=0$. Here the counter example of all ...
1
vote
1answer
2k views

Fourier Sine Transform of the derivative of a function [closed]

I'm going over some old stuff of Fourier transforms, and came across the identity $\mathscr{F}_s[f']=-\omega\mathscr{F}_c[f].$ I know this is done using integration by parts but I'm having a problem ...
4
votes
2answers
501 views

Question about proof of Fourier transform of derivative

If $f\in L^1(\mathbb{R})$, $f'(x)$ exists and is continuous, and $f'\in L^1(\mathbb{R})$, then $\widehat{f'}(t)=2\pi i \widehat{f}(t)$. I've stated the above theorem from a textbook that I'm reading. ...
1
vote
2answers
245 views

The Fourier transform of the derivative of a function $f\in L_1(\mathbb{R})$

Let $f\in L_1(\mathbb{R})$ (That is to say $f$ is absolutely integrable over $\mathbb{R}$) with derivative $f'\in L_1(\mathbb{R})$. The Fourier transform of $f$ is given by: \begin{align} \hat{f}(t) =...
1
vote
1answer
238 views

Fourier transform of derivative when integrating by parts

As seen in https://math.stackexchange.com/a/430885/634773, we can find the Fourier Transform of the derivative of a function through the anti-transform. But if we do integration by parts, shouldn't ...
2
votes
0answers
274 views

Fourier Transform of derivative a vector function

I am working on the proof of the Fourier Transform of the derivative of a function. I am accompanied by some proof lines but having some issue in one of the integral evaluation. I searched out its ...
1
vote
1answer
124 views

Application of the operator $\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right)$

I need to apply the operator $$\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right) \tag{1} \label{1}$$ To the function $$M(x) N(y +C_{1}p)\mathcal{F}[f(q)](p) \tag{2} \label{2}$$ where $M(x)$ ...
1
vote
1answer
100 views

How do I find the Fourier transform of $\mathcal{F}[\log(a^2+s^2)](s)$

For $a>0$ i have managed to show that this is the Fourier transform of the function. $$ \mathcal{F}[e^{-a|x|}](s) = \frac {2a}{\sqrt{2{\pi}}(a^2+s^2)}. $$ How do I now use this to find the ...
1
vote
0answers
56 views

Fourier Transform of a derivative - Integration

I have been trying to understand the formula for the Fourier Transform of a derivative, but I am getting hung up in the integration. Looking at this post. So far, I understand everything up to the ...
2
votes
0answers
56 views

Fourier transform on Green's function

I need to do a Fourier transform for the next Green's function:$F[(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}-m^{2})\cdot G(x,x^{'})]$. My solution is: $\int_{-\infty }^{\infty }(\frac{\mathrm{d} ^{2}}{...
1
vote
1answer
56 views

Fourier transform of the following time dependent expression

I am working with the following expression which describes a simple Markov birth-death model of particle transport of uniform diameter: \begin{gather*} \frac{\partial\eta(t)}{\partial t}=\nu+\mu\eta-\...
0
votes
0answers
31 views

Fourier transformation of this equation

The function I have is f(x) = x * $\exp(-x^2)$ but the problem is I don't have any domain of integration for it so i assume you take (-$\infty$,$\infty$) ... As follows I test the party of the ...