Fourier transform in <PERSON>'s paper
While calculating $\beta_{ij}$ for the case of a Schwarzschild black hole, <PERSON> uses the Fourier transform of the solution of the wave equation (Particle Creation by Black Holes, <PERSON>, Commun. Math. Phys. 43, 199 (1975)).
So the first question I have is this: Is the reason why he prefers to work in Fourier space that in there the differential equation becomes an algebraic equation, making it easier to deal with?
The other question pertains to the mathematical machinery of the transform. 346
The wave equation is $\Box \phi=0$, and it can be expanded in terms of the complete orthonormal set ${{g}}$ satisfying $\Box g=0$. In Schwarzschild spacetime we also have another additional property, i.e. $g(r,t,\theta,\phi)=\frac{1}{r}f(r,t)Y_{lm}(\theta,\phi)$, where $Y_{lm}(\theta,\phi)$ is the spherical harmonic function and $f$ satisfies the following equation $$\frac{\partial^2 f}{\partial t^2}-\frac{\partial^2f}{\partial r_}+\left(1-\frac{2M}{r}\right)\left[\frac{l(l+1)}{r^2}+\frac{2M}{r^3}+m^2 \right]f=0 \tag{1}$$ where $r_=r+2Mln(\frac{2M}{r}-1)$.
Now, <PERSON> states that Fourier transform of the incoming wave can be written as the following: $$p_{\omega^{'}lm}=\frac{1}{\sqrt{2\pi\omega^{'}}}\times\frac{1}{r}\times F_{\omega^{'}}(r)\times e^{i\omega^{'}v}\times Y_{lm}(\theta,\phi) \tag{2} $$ where $v=t+r_*$.
Going by the definition of the Fourier transform, one obtains $$p_{\omega^{'}lm}=\int_{-\infty}^{+\infty}\frac{1}{r}f(r,t)Y_{lm}(\theta,\phi)e^{-i\omega^{'}v}dv \tag{3}$$ since $dv=dt+(1-\frac{2M}{r})^{-1}dr$. I don't have any idea how to proceed any further from here, so what should I do now?
Edit: My problem is how to get the final form of $p_{\omega^{'}lm}$ and by final form I mean the equation 2.12 or 2.11
And if anyone is still confused as what one has to show it's just how one goes from $(3)$ to $(2)$. 804