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I need to do a Fourier transform for the next Green's function:$F[(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}-m^{2})\cdot G(x,x^{'})]$.

My solution is: $\int_{-\infty }^{\infty }(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}\cdot G(x,x')\cdot e^{-ikx})dx-m^{2}\int_{-\infty }^{\infty }G(x,x')\cdot e^{-ikx}dx=\int_{-\infty }^{\infty }(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}\cdot G(x,x')\cdot e^{-ikx})dx-m^{2}\widetilde{G}(k)$

but I don't know how to solve the first integral.

The solution is:$-(k^{2}+m^{2})\cdot \widetilde{G}(k)$

Can someone please explain to me how did they get this expression?

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  • 1
    $\begingroup$ Looks related to math.stackexchange.com/questions/430858/… $\endgroup$ – Peter Jan 10 '20 at 9:06
  • $\begingroup$ Use partial integration twice and then that G(x,x') should vanish for x to $\pm \infty$ $\endgroup$ – Peter Jan 10 '20 at 9:09
  • $\begingroup$ Thank you all for the help, I have completely forgotten about differentiation property of the Fourier transform. $\endgroup$ – violettagold Jan 10 '20 at 9:12

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