Confirm by direct substitution into the defining equation that the free-space Green function for the Helmholtz equation

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Confirm by direct substitution into the defining equation that the free-space Green function for the Helmholtz equation in two-dimensional plane polar coordinates ρ = (ρ,φ) isimage

where H(1) 0 (x) is the zero-order Hankel function of the first kind.

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