Greens function has me blue ....

  • #1
ognik
643
2
Not following this example (PDE for Greens function) in my book:

Book states:

I recognised this as the Hemlmholtz eqtn, but cannot find where the 3rd term comes from? It looks like it could be the 3D Fourier Transform representation of the dirac-delta function? (if so a link to, or a derivation would be nice)

Then they say they solve the PDE in terms of a Fourier Integral

I know the Fourier Integral in 3D is , (sqrt in denominator?) so I'm not sure what they are doing here?
 
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  • #2
The third term in your example is indeed the 3D Fourier transform of the Dirac delta function. The 3D Fourier transform of a function is given by where . This expression can be thought of as a generalization of the 2D Fourier transform, which uses a square root in the denominator instead of a cube root.The solution to the PDE that they give is a general solution to the homogeneous equation , which can be written asThis is just the 3D Fourier transform of a function , which is some arbitrary function of momentum .
 

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