This problem outlines a contour integration method to prove that See Problem 8.23 for another method. (a)

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This problem outlines a contour integration method to prove that

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See Problem 8.23 for another method.

(a) The left side of the Weyl identity is the free-space Green function, G0(r), which satisfies (∇2 + k20)G0(r) = −δ(r). Fourier transform this differential equation and show that

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(b) Use contour integration to perform the integral over kz in part (a). Assume that k0 has a small positive imaginary part to establish the location of the poles and to decide how to close the contour.


Data From Problem 8.23

Write δ(r − r') = δ(r⊥ − r')δ(z − z') and use direct integration to derive Weyl’s formula for the free-space Green function in three dimensions,

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