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I wonder if there is anybody who can give me a hint on how to handle the following situation:

In the 2+1 dimensional Klein-Gordon equation with coordinates (t,x,y), I use as initial condition for [tex]\Psi(x,0)[/tex] a spherically symmetric Gaussian. The relativistic dispersion relation is of course [tex]\omega^2=k_x^2+k_y^2[/tex].

I can now Fourier transform [tex]\Psi(x,0)[/tex] to [tex]\Phi(k,0)[/tex], no problem.

But when I want to calculate [tex]\Psi(x,t)[/tex], by doing a second Fourier transformation, I get stuck because of the square root of the dispersion relation.

In other words, I can't solve the integral [tex]\int_{-\infty}^{+\infty} dk_x dk_y \exp\left( i k_x x+i k_y y - i \sqrt{k_x^2+k_y^2}t -\alpha^2(k_x^2+k_y^2)/2\right) [/tex].

If anybody could give me a hint on this, I would be very thankful and happy, I already spent way too much time on this!

Thanks very much!!