Krdytkyu

I'm following a textbook in which they introduce

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

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

Using the Fourier Transform convention

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

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

which shouldn't be a surprise.