I'm Solving (or at least trying to solve) 4 Klein Gordon equation with sources (Maxwell equations) with the Green function method, therefore I have
$$ \Box G(x,x')= \delta^4(x-x') \tag{1}$$
Where $G$ is the green function and the arguments of the functions are 4-vectors.
To Find the Green function we use the Fourier transform, and here there is my problem, which is mainly a terminology problem. We write
$$ G(x-x') = \frac{1}{(2 \pi)^4} \int d^4k \tilde{G}(k) e^{-ik \cdot (x-x')} \tag{2}$$
Now to my definition of Fourier Transform which is $$ (F f)(x)= \tilde{f}(x)= \frac{1}{(2 \pi)^{n/2}} \int \,\tilde{f}(k) e^{-ix \cdot k} d^nk \tag{3}$$
I have that $G$ is the Fourier transform of $\tilde{G}$, apart from a factor $(2\pi)^{1/2}$ missing in equation $(2)$ which i cannot explain.
Is my view right? Both my book and my notes says that $\tilde{G}$ is the Fourier transform of $G$ while in my opinion $\tilde{G}$ is the inverse Fourier transform of $G$.
I mean in $(2)$ I'm Fourier transforming the inverse Fourier transform of $G$ and that's why I obtain $G$, which is by means of the inversion equation: $$ f=(F(\overline{F} f) \tag{4}$$
Where $\overline{F} f$ is the inverse Fourier Transform.
I'm probably missing something and the answer may be banal, but, if I am, where am I wrong? And why is there a missing factor of $(2\pi)^{1/2}$ ?
Hoping that someone enlighten me, thanks in advance for any help.
