Linked Questions

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}^{...

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 ...

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 ...

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. ...

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) =...

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 ...

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 ...

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 $M(x)$ ...

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 ...

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 ...

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}}{...

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-\...

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 ...