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Then using the expression for Legendre's polynomials in terms of the hyper-geometric function (Bateman and Erdelyi, M1953)

and representing the series in Eq. 2.14.29 in the form of Neumann's expansion (see Watson, M1949) we obtain the closed expression for the function :

Substituting Eq. 2.14.31 in Eq. 2.14.26 we derive the expression for the Green's function G(r, ro ;w) in the form of contour integral

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