Inverse fourier transform for function with three variables

Question

I'm following a textbook in which they introduce

$V_{p,q}(\omega,\tau,z)= \frac{1}{2\pi} \int e^{-ih(\tau-(p+q)z/c)}U_{p,q}(\omega,h,z)dh$

where $p$ and $q$ are integers. How do I find an expression for $U_{p,q}$?

Answer 1

Using the Fourier Transform convention

$$\mathscr{F}\left\{f(\tau)\right\} = \int_{-\infty}^{\infty} f(\tau)e^{ih\tau} d\tau = \hat{f}(h)$$ $$\mathscr{F}^{-1}\left\{\hat{f}(h)\right\} = \dfrac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(h)e^{-ih\tau} dh = f(\tau)$$ Then

$$\begin{align*}V_{p,q}\left(\omega,\tau,z\right)&= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-ih(\tau - (p+q)z/c)}U_{p,q}\left(\omega,h,z\right)dh\\ \\ &= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-ih\tau}e^{ih\frac{(p+q)z}{c}}U_{p,q}\left(\omega,h,z\right)dh\\ \\ &= \mathscr{F}^{-1}\left\{e^{ih\frac{(p+q)z}{c}}U_{p,q}\left(\omega,h,z\right)\right\}\\ \\ \mathscr{F}\left\{V_{p,q}\left(\omega,\tau,z\right)\right\} &=e^{ih\frac{(p+q)z}{c}}U_{p,q}\left(\omega,h,z\right) \\ \\ e^{-ih\frac{(p+q)z}{c}}\mathscr{F}\left\{V_{p,q}\left(\omega,\tau,z\right)\right\} &=U_{p,q}\left(\omega,h,z\right) \\ \\ e^{-ih\frac{(p+q)z}{c}}\int_{-\infty}^{\infty}e^{ih\tau}V_{p,q}\left(\omega,\tau,z\right)d\tau &=U_{p,q}\left(\omega,h,z\right) \\ \\\int_{-\infty}^{\infty}e^{ih\left(\tau-\frac{(p+q)z}{c}\right)}V_{p,q}\left(\omega,\tau,z\right)d\tau &=U_{p,q}\left(\omega,h,z\right) \\ \end{align*}$$

which shouldn't be a surprise.