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- Homework Statement
- Help with evaluating Fourier transform

- Relevant Equations
- The definition of Fourier transform (F.T.) that I am using is given as:

$$f(\vec{x},t)=\frac{1}{\sqrt{2\pi}}\int e^{-i\omega t}\tilde{f}(\vec{x},\omega)\,\mathrm{d}\omega$$

The definition of Fourier transform (F.T.) that I am using is given as:

$$f(\vec{x},t)=\frac{1}{\sqrt{2\pi}}\int e^{-i\omega t}\tilde{f}(\vec{x},\omega)\,\mathrm{d}\omega$$

I want to show that:

$$\frac{1}{c\sqrt{2\pi}}\int e^{-i\omega t}\omega^2 e^{ikx}(\vec{x}\times\vec{p}_{\omega})\,\mathrm{d}\omega=-\frac{\vec{x}}{c}\times\frac{d^2}{dt^2}\,\vec{p}(t-x/c)$$

To show the above, two F.T. must occur to get the second derivative w.r.t. time and the ##(t-x/c)## term. However, I am not sure how to proceed since obtaining either one will induce another F.T. into the second equation. How should I continue?

$$f(\vec{x},t)=\frac{1}{\sqrt{2\pi}}\int e^{-i\omega t}\tilde{f}(\vec{x},\omega)\,\mathrm{d}\omega$$

I want to show that:

$$\frac{1}{c\sqrt{2\pi}}\int e^{-i\omega t}\omega^2 e^{ikx}(\vec{x}\times\vec{p}_{\omega})\,\mathrm{d}\omega=-\frac{\vec{x}}{c}\times\frac{d^2}{dt^2}\,\vec{p}(t-x/c)$$

To show the above, two F.T. must occur to get the second derivative w.r.t. time and the ##(t-x/c)## term. However, I am not sure how to proceed since obtaining either one will induce another F.T. into the second equation. How should I continue?

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