Order of multi-variable integration of infinite range

[cutemango3]

Homework Statement

If the Green's function of the electric field in a system is

[tex] G(x,x')=e^{-i(x-x')^2}[/tex]

I want to calculate the phase of the electric field at [itex] x [/itex] if the source is uniformly distributed at [itex]x'=-\infty [/itex] to [itex]x'=\infty[/itex]

Homework Equations

The Attempt at a Solution

Then, the phase of the field at [itex]x[/itex] may be

[tex] \theta_1\equiv Arg\Big[\int_{-\infty}^{\infty}e^{-i(x-x')^2}dx'\Big]=Arg\Big[\int_{-\infty}^{\infty}e^{-ix'^2}dx'\Big]=Arg[1-i]=-\frac{\pi}{4}[/tex]

The phase of the field may be independent from [itex] x[/itex]. So, I calculate the following:
[tex]\theta_2\equiv Arg \Big[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-i(x-x')^2}dxdx' \Big] =Arg \Big[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-i(x^2-2xx'+x'^2)}dxdx' \Big]=Arg \Big[ \int_{0}^{\infty}\int_{0}^{2\pi}e^{-ir^2(1-\sin 2\phi)}d\phi dr \Big]
=Arg \Big[ 2\pi \int_{0}^{\infty} e^{-ir^2}J_0(r^2) dr \Big]
[/tex]

[itex]\theta_1 =\theta_2 [/itex]? I think that that should be the case.

 

[Nino MMP]

Then, the phase of the electric field at x if the source is uniformly distributed at x'=-\infty to x'=\infty should be \theta_1 =-\frac{\pi}{4}.Is my solution correct?Thanks in advance !