So, in the calculation of D(t,r)=[ϕ(x),ϕ(y)], where t=x0−y0, →r=→x−→y you need to calculate the following integral D(t,r)=12π2r∞∫0dppsin(pr)sin[(p2+m2)1/2t](p2+m2)1/2
For m=0, the integral is simple. We get D(t,r)=14πr[δ(t−r)−δ(t+r)]
I even know what the answer for m≠0. I have no idea how to calculate it though. Any help?
Answer
Using Gradshteyn and Ryzhik (seventh edition) 3.876 (1) $$\int_0^\infty \frac{\sin{(p \sqrt{x^2+a^2})}}{\sqrt{x^2+a^2}} \cos(b~x)dx=\frac{\pi}{2} J_0(a\sqrt{p^2-b^2}) ~~[0
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