The plane wave decomposition is mathematically universal??? 1. My questions is: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves? Note: r=0 is a singularity. This is actually the potential function produced by an ideal radiation electric dipole. [J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), Chapter 9, p. 410, Eq. (9.16).] (i) A spherical-wave decomposition of a plane wave is presented in textbook; for example, J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), p. 471, Eq. (10.44) in Chapter 10. (ii) The converse: A plane-wave decomposition of spherical waves is given by MacPhie and Ke-Li Wu, “A Plane Wave Expansion of Spherical Wave Functions for Modal Analysis of Guided Wave Structures and Scatterers”, IEEE Trans. Antennas and Propagation 51, 2801 (2003). Note: The spherical waves are analytical at r=0. 2. Where can I find the conclusions: (a) The plane wave decomposition is mathematically universal. (b) Any spherical wave may be decomposed in a plane wave basis. Are they math theorems? Thanks a lot.